# Assume that x and y are both differentiable functions of t and find the required values of...

## Question:

Assume that {eq}x{/eq} and {eq}y{/eq} are both differentiable functions of {eq}t{/eq} and find the required values of {eq}\frac{dy}{dt}{/eq} and {eq}\frac{dx}{dt}{/eq}.

(a) Find {eq}\frac{dy}{dt}{/eq} when {eq}x = 5{/eq}, given that {eq}\frac{dx}{dt} = 3{/eq}.

{eq}\frac{dy}{dt} = {/eq}

(b) Find {eq}\frac{dx}{dt}{/eq} when {eq}x = 8{/eq}, given that {eq}\frac{dy}{dt} = 2 {/eq}.

{eq}\frac{dx}{dt} ={/eq}

MISSING INFORMATION

## Derivatives:

For the parametric functions {eq}x=f(t), \ y=g(t)
{/eq}, we can calculate the slope of the tangent {eq}\dfrac{dy}{dx}
{/eq} as {eq}\boxed{\dfrac{\frac{dy}{dt}}{\frac{dx}{dt}}=\dfrac{dy}{dx}}
{/eq}. Derivatives of the functions represent the slope of the tangent to the curves at a given point.

## Answer and Explanation: 1

Become a Study.com member to unlock this answer! Create your account

View this answer

Let {eq}y=x^2
{/eq}

a) Given:

- {eq}\dfrac{dx}{dt} = 3 {/eq}

- {eq}x=5, \dfrac{dy}{dx}_{ (at \ x=5)}=2(5)=10 {/eq}

We know that...

See full answer below.

#### Ask a question

Our experts can answer your tough homework and study questions.

Ask a question Ask a question#### Search Answers

#### Learn more about this topic:

from

Chapter 7 / Lesson 5The derivative in calculus is the rate of change of a function. In this lesson, explore this definition in greater depth and learn how to write derivatives.

#### Related to this Question

- Assume that x and y are both differentiable functions of t , and find the required values of d y d t and d x d t . y = 2 ( x 2 ? 3 x ) (a) Find d y d t when x = 3 , given that d x d t = 5
- Assume that x and y are both differentiable functions of t and find the required values of \frac{dy}{dt} and \frac{dx}{dt} y=\sqrt x (a) Find \frac{dx}{dt}, given x = 25 and \frac{dy}{dt} = 3. (
- Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. xy = 4 a) Find dy/dt, given x = 8 and dx/dt = 11.(b) Find dx/dt, given x = 1 and dy/dt=-8.
- Assume that x and y are both differentiable functions of t and find the required values of dy / dt and dx / dt. xy = 8. (a) Find dy / dt, given x = 6 and dx / dt = 13. (b) Find dx / dt, given x = 1 a
- 1. Assume that x and y are both differentiable functions of t and find the required values of d y d t and d x d t . (a) Find d y d t when x = 5, given that d x d t = 5. (b) Find d x d t whe
- Given x^2 + y^2 = 225. Assume that x and y are both differentiable functions of t and find the required values of dx/dt. Find dy/dt, given x = 9, y = 12, and dx/dt = 8. Find dx/dt, given x = 12, y = 9, and dy/dt = 4.
- Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. xy = 4 (a) Find dy/dt, given x = 2 and dx/dt = 15. (b) Find dx/dt, given x = 1 and dy/dt = -6.
- Assume that x and y are both differentiable functions of t and find the required values of d y d t , d x d t , x y = 4. a) find d y d t , given that x=2 and d x d t = 14. , b) find d x d t given x=1 and d y d t = 6
- Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. xy = 2 a. Find dy/dt, when x = 4, given that dx/dt = 13. b. Find dx/dt, when x = 1, given that dy/dt = -9.
- Assume that \, x\, and \, y\, are both differentiable functions of \, t\, and find the required values of \, dy/dt\, and \, dx/dt. \begin{array}{cclcc} \underline{\text{Equation & \hspace{0.20in} & \hfill\underline{\text{Find\hfill & \hspace{0.20
- Assume that \, x\, and \, y\, are both differentiable functions of \, t\, and find the required values of \, dy/dt\, and \, dx/dt. \begin{array}{cclcc} \underline{\text{Equation & \hspace{.19in} & \hfill\underline{\text{Find\hfill & \hspace{.19in
- 1. Assume that x and y are both differentiable functions of t, find the required values of dy/dt and dx/dt. xy = 4 (a) Find dy/dt, when x = 2, given that dx/dt = 10. (b) Find dx/dt, when x = 1, given
- Assume that \, x\, and \, y\, are both differentiable functions of \, t\, and find the required values of \, dy/dt\, and \, dx/dt. \begin{array}{cclcc} \underline{\text{Equation & \hspace{0.19in} & \hfill\underline{\text{Find\hfill & \hspace{0.19
- Assume that \, x\, and \, y\, are both differentiable functions of \, t\, and find the required values of \, dy/dt\, and \, dx/dt. \begin{array}{cclcc} \underline{\text{Equation & \hspace{.20in} & \hfill\underline{\text{Find\hfill & \hspace{.20in
- Assume that x and y are both differentiable functions of the required value of dy/dt and dx/dt. xy=8 Find dx/dt, when x=1, given that dy/dt=-7.
- Assume that x and y are both differentiable functions of the required value of dy/dt and dx/dt. xy=8 Find dy/dt, when x=6, given that dx/dt=12.
- Assume that x and y are differentiable functions of t . Find \frac{\text{d}y}{\text{d}t}(4) using the given values: xy=x+4,\ \frac{\text{d}x}{\text{d}t}(4)= 5 .
- Assume x and y are both differentiable functions of t and 3x^yy = 30 . Find \frac{dy}{dt} if \frac{dx}{dt} = 6 and x=1 .
- Assume that x and y are differentiable functions of t. For the following, find \frac{dx}{dt} given that x = 3, y = 9, and \frac{dy}{dt} = 2. y^2 = 2xy + 27
- Find the values of m and b that make the following function differentiable; f(x) = \left\{ {\matrix{ x^3 \cr {mx + b} \cr } } \right.~~\matrix{ {x \le 1} \cr {x greater than 1} \cr }
- Suppose f and g are functions that are differentiable at x = 1 and that f(1) = 2, f'(1) = -1, g(1) = -2, and g'(1) = 3. Find the value of h'(1). h(x) = (x^2 + 4) g(x)
- Suppose f and g are functions that are differentiable at x = 1 and that f(1) = 2, f'(1) = -1, g(1) = -2, and g'(1) = 3. Find the value of h'(1), where h(x) = (x f(x)) / (x + g(x)).
- Assume that x and y are both differentiable functions of the required value of dy/dt and dx/dt. y=square root of{x} Find dy/dt, when x=9, given that dx/dt=3.
- Assume that x and y are both differentiable functions of the required value of dy/dt and dx/dt. y=square root of{x} Find dx/dt, when x=81, given that dy/dt=3.
- Suppose that f(x) and g(x) are differentiable functions such that f(1) = 2, f'(1) = 4, g(1)=5,\space \text{and} g' (1)= 3. Find h' (1) when h(x)=\frac{f(x)}{g(x)}.
- Suppose that f and g are functions that are differentiable at x = 1 and that f(1) = 2, f '(1) = -1, g(1) = -2 , and g'(1)=3 . Find h'(1). h(x) = ( x^2 + 8 ) g(x)
- Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. y = ? x (a) find dy/dt, given x=16 and dx/dt=2 dy/dt= (b) find dx/dt,given x=36 and dy/d
- Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find
- Show how to find all values at which a function is differentiable.
- Suppose u and v are differentiable functions of x and that u(0) = 5, u'(0) = -3, v(0) = -1, v'(0) = 2. Find the value of d(7v - 2u)/dx at x = 1.
- Assume that (f/g) (x) = x^2 + 2x, where f and g are differentiable functions such that f (2) = 2 and f' (2)= 3. Find g' (2).
- Suppose f \text{ and } g are functions that are differentiable at x = 1 and that f(1) = 2, \ f (1) = 1, \ g(1) = 2, \text{ and } g (1) = 3 . Find the value of h (1) , where h(x) = (x^2 + 11) g(x) .
- Suppose f and g are functions that are differentiable at x = 2 and that f' (2) = 1, f (2) = 3, g' (2) = -1 and g (2) = 2. Then find the value of h' (2) if h (x) = x. f (x).
- Assume that f(x) and g(x) are differentiable for all x. Let h(x) = 2f(x) + \frac{g(x)}{7}, find h'(x).
- Assume that f(x) and g(x) are differentiable for all x. Let \displaystyle h(x) =\frac{f(x)g(x)}{5}, find h'(x). h'(x) = _____
- Suppose y = f(x) is differentiable function of x that satisfies the equation x^2y+y^2 = x^3. Find dy/dx implicitly
- Given that s^2 = x^2 + y^2, where x and y and s are differentiable function of t, find \frac{ds}{dt} when s = 20, x = 15, and \frac{dy}{dt} = 4 and \frac{dx}{dt} = 5.
- Find the value of a that makes the following function differentiable for all x-values. g(x)={ ax,&{if}&x 0x^2-9x,& {if}&xgeq 0.
- Let f be a differentiable function. If h (x) = (1 + f (3x))^2, find h' (x).
- (a) Suppose that f and g are function that are differentiable at x = 1 and that f (1) = 4, f' (1) = -6, g (1) = -4, and g' (1) = 2. Find h'(1). h (x) = {f (x) g (x)} / {f (x) - g (x)}. (b) Find the de
- Suppose that f and g are functions that are differentiable at X 1 and that f(1) 2, f (1) -1, g(1) -2, and g (1) 3. Find h 1). h(x) (x2 10)g(x) h(1)
- Suppose that f and g are functions that are differentiable at x = 1 and that f (1) = 2, f' (1) = -1, g (1) = -2, and g' (1) = 3. Find h' (1). h (x) = (x^2 + 7) g (x).
- Suppose that f and g are functions that are differentiable at x = 1 and that f(1) = 4, f '(1) = -4, g(1) = -4, and g'(1) = 4. Find h'(1). h(x) = f(x)g(x) / (f(x) - g(x))
- Suppose that f and g are functions that are differentiable at x = 1 and that f (1) = 2, f' (1) = -1, g(1) = -2, and g'(1) = 3. Find h' (1). h (x) = (x^2 + 9) g (x)
- Suppose that f and g are functions that are differentiable at x = 1 and that f(1) = 4, f '(1) = - 6, g(1) = - 4, and g'(1) = 2. Find h'(1). h(x) = (f(x) g(x)) / (f(x) - g(x))
- Suppose that f is a differentiable function with f_x (8,0) = 8 and f_y (8, 0) = 7. Let w(u, v) = f (x(u, v), y(u, v)) where x = 8 cos u + 2 sin v and y = 8 cos u sin v. Find w_v (0, 0).
- Let f(x, y, z) be a given differentiable function and define a new function g(x, y, z) = f(yz, zx, xy). Suppose that f_{x}(1, 1, 1) = 1, f_{y}(1, 1, 1) = 2, f_{z}(1, 1, 1) = 3. Find: a) g_{x}(1, 1, 1
- Let f (x, y, z) be a given differentiable function and define a new function g (x, y, z) = f (yz, zx, xy). Suppose that f_x (1, 1, 1) = 1, f_y (1, 1, 1) = 2, f_z (1, 1, 1) = 3. Find the following. (a)
- Let f(x,y,z) be a given differentiable function and define a new function g(x,y,z) = f(yz,zx,xy). suppose that f_x(1,1,1) = 1 , f_y(1,1,1) = 2 , f_z(1,1,1) = 3 . Find . (a) g_x(1,1,1) . (b) g_y(1,1,1)
- Find the values of a and b that make the following function differentiable for all x \\ f(x) = \begin{cases} a^2x + b, & \text{ if } x \gt 1\\ bx^2 + 3, & \text{ if } x \leq 1 \end{cases}
- The function f is differentiable and \int_0^x (4f(t)+5t)dt=\sin (x). Determine the value of f'(\frac{\pi}{6}). a. \frac{5}{4} b. \frac{\sqrt 3}{8}-\frac{5\pi}{24} c. 0 d. \frac{\sqrt 3}{2} e. -\frac{11}{8}
- Find the values of a and b that make the following function differentiable for all x-values. f(x) = \begin{cases} ax + b, x -1 \\bx^2 - 3, x\leq -1\end{cases}
- Suppose that f(x) and g(x) are differentiable functions such that f(5) = 4, f'(5) = 3, g(5) = 2, and g'(5) = 1. Find h'(5) when h(x) = f(x)/g(x).
- Suppose that f(x) and g(x) are differentiable functions such that f(0) = 9, \enspace f'(0) = 7, \enspace g(0) = 4, \enspace g'(0) = 2. Find \dfrac{d}{dx} \left(f(x)g(x)\right) \Bigg|_{x = 0}.
- Assume that x and y are both differentiable functions of t and find the required values of \frac{dx}{dt}. x^{2} + y^{2} = 25 (a) Find \frac{dy}{dt}, given x = 3, y = 4, and \frac{dx}{dt} = 7. (b) Find
- Find the values of a and b that make the following function differentiable for all x values. f(x) = { a(2x+4/x^2+1) + be^{sin x} if x greater than equal to 0, x^4+x+7, if x less than 0.
- Suppose that f(x) and g(x) are differentiable such that f(2) = 6, f '(2) = 4, g(2) = 7, and g '(2) = 1. Find h '(2) when h(x) = f(x) / g(x). h '(x) =
- Suppose f is differentiable on R. Let F(x) = f(e^x) and G(x) = e^f(x). Find expressions for (a) F'(x) and (b) G'(x).
- Assuming that this function is differentiable, solve y = \frac{1 + xf(x)}{\sqrt {x.
- Suppose f and g are functions that are differentiable at x= 1 and that f(1)= 7, f '(1)= 9, g(1)= 6, and g '(1)= 2. Find the value of h '(1). h(x)= \frac{f(x)g(x)}{f(x) - g(x)}
- Suppose that f and g are functions that are differentiable at x = 1 and that f(1) = 2, f'(1) = -1, g(1) = -2, g'(1) = 3. Suppose h(x) = f(x)g(x). Find the value of h'(1).
- Find: Determine for what points the following functions are differentiable and describe (qualitatively) 1. f (x,y) = e^{xy} \cos (\pi(xy+1)) 2. g (x,y) = \frac{x4 - y4 }{ x + y} 3. h (x,y) = x - 2y \
- Suppose that f\left ( x \right ) and g\left ( x \right ) are differentiable functions such that f\left ( 6 \right ) = 1, {f}' \left ( 6 \right ) = 9, g\left ( 6 \right ) = 8 and {g}' \left ( 6 \right ) = 3. Find {h}' \left ( 6 \right ) when h\left ( x \ri
- Show that f ( x ) = { x 2 sin ( 1 x ) , i f x 0 0 , i f x = 0 is differentiable at x = 0 and find f ( 0 )
- Suppose f is differentiable on R. Let F(x) = f(ex) and G(x) = ef(x). Find expressions for (a) F (x) and (b) G (x)
- Suppose f is a differentiable function of x and y, and g(r,s)=f(5r-s,s^{2}-6r) . Use the table of values below to calculate g_{r} (9,4) and g_{s} (9,4)g_{r}(9,4)= g_{s}(9,4)=
- Suppose f is a differentiable function of x and y, and g(r,s)=f(4r-s, \ s^2-7r). Calculate g_r(4,1) and g_s(4,1) using the given table of values. \displaystyle \begin{array}{|c|c|c|c|c|} \hline
- Let p ( t ) = f ( g ( t ) , h ( t ) ) , where f is differentiable, g ( 2 ) = 4 , g ? ( 2 ) = ? 3 , h ( 2 ) = 5 , h ? ( 2 ) = 6 , f x ( 4 , 5 ) = 2 , f y ( 4 , 5 ) = 8. . Find p ? ( 2 ) .
- If f(x) is a function differentiable at x = 1 and f(1) = 1/8. What is the value of f(x) - f(1)
- Find A and B given that the function, f(x) is differentiable at x=1. f(x)= \begin{cases} x^2-5 & x<1 \\ Ax^2+Bx & x \geq1 \end {cases}
- Suppose that f and g are functions that are differentiable at x = 1 and that f(1) = 1, \; f'(1) = -3, \; g(1) = 2, and g'(1) = 5. If h(x)=f(x)g(x), find h'(1).
- a) Find all functions f such that f'(x) = 1 for all x. b) For a differentiable function g, find an expression for(1/g)' (x). c) Let h be a differentiable function such that h'(x) = h^2(x) for all x (this is known as a differential equation). Find (1/h)'
- Find f' for each of the following where g and k are differentiable functions. F= g(\sqrt{k(x^2)})
- Use the table of values for the differentiable functions f(x) and g(x) to evaluate the following expressions: \begin{array}{|l|l|l|l|l|} \hline x & f(x)&g(x)&f'(x)&g'(x)\\ \hline -3&0&-3&5&3\\ \hline
- Suppose f is a differentiable function such that (1) f(x + y) = f(x) + f(y) + 2xy for all real numbers x and y and (2) \lim_{h \to 0} \frac{f(h)}{h} = 7. Determine f(0) and find the expression for f(x). (Hint: find f'(x).)
- Suppose h and g are functions that are differentiable at x = 1 and that f(1) = 2, f'(1) = -1, g(1) = -2 and g'(1) = 3. Find the value of h'(1). h(x) = \frac{x f(x)}{x + g(x)}. The answer should be
- Suppose u and v are differentiable functions at t = 0 with u(0) = <0,1,1>, u'=<0,7,1>, v(0) = <0,1,1>, and v'(0) = <1,1,2>. Evaluate: a) \frac {d}{dt} (u.v)|_{t=0}
- Find h (x) where f(x) is an unspecified differentiable function. h(x) = f(x) / x^{15}
- Find f'(x) for the following functions. (a) f(x) = \sin^4(4x^5) (b) f(x) = g^n(\tan(x^2)), where g is a differentiable function and n is a constant.
- Suppose u and v are differentiable functions of x and that u(0) = 5, u'(0) = -3, v(0) = -1, v'(0) = 2. Find the value of d(u/v)/dx at x = 1.
- Suppose u and v are differentiable functions of x and that u(0) = 5, u'(0) = -3, v(0) = -1, v'(0) = 2. Find the value of d(v/u)/dx at x = 1.
- Suppose u and v are differentiable functions of x and that u(0) = 5, u'(0) = -3, v(0) = -1, v'(0) = 2. Find the value of d(uv)/dx at x = 1.
- Let f(x, y, z) be a given differentiable function and define a new function g(x, y, z) = f(yz, zx, xy). Suppose that f_x(1, 1, 1) = 1, f_y(1, 1, 1) = 2, f_z(1, 1, 1) = 3. Find the following.g_x
- Find: Suppose f is a differentiable function such that 1. f(g(x)) = x and f (x) = 1 + \left [ f(x) \right ] 2. Show that g (x) = \frac{1}{(1 + x^2)}.
- Suppose that f(x) \text{ and } g(x) are differentiable functions such that f(6)=2, \ f (6)=3, \ g(6)=8, \text{ and } g (6)=9 . Find \frac{\text{d{\text{d}x} \left(f(x)g(x)\right) \text{ at } x=6 .
- (a) Consider the function f(x, y) = xy, where x = u(t) and y = v(t) (here u and v are differentiable functions of t). Compute \frac{df}{dt}. The result should look familiar. (b) Now let f(x, y) =
- Two functions f & g are differentiable and h(x)= f(g(x)). Suppose f(0)=4, f'(0)=3, g(0)=1, g'(0)=2, h(0)=1, h'(0)=6. Find the values of f(1) and f'(1).
- If z = f(x, y), where f is differentiable, and x = g(t) , y = h(t) g(4) = 5 , h(4) = -5 g'(4) = -5, h'(4) = -1 { f_x (5, -5) = 1, f_y (5, -5) = 6 } find dz/dt when t = 4. { \frac{dz}{dt} =
- The differentiable function f satisfies f(-2) = 0 , f'(-2) = 0, f(0) = 4, f'(0) = -2, f(3) = 1, and f'(3) =-3. Evaluate \int_{-2}^{3}4(f(z)+1)^{2}f'(z)dz.
- Suppose that the differentiable function y = f(x) has an inverse and that the graph of passes through the point (2, 4) and has a slope of \frac{1}{3} there. Find the value of \frac{df^{-1{dx} at x = 4.
- Suppose f is a differentiable function of x and y, and g(r,s)=f(5r-s, s^2-9r). Use the table to calculate g_r(9,8) and g_s(9,8). &f&g&f_x&f_y (37,-17)&5&1&3&8 (9,8)&1&5&6&2
- Suppose f is a differentiable function of x and y , and g(r,s) = f(5r-s, s^2-9r) . Use the table below to calculate g_r(9,8) and g_s(9,8) | | f | g | f_x | f_y | (37,-17) | 5 | 1 | 3 |
- Find the function below, find the values of x in which f'(x) = 0. f(x) = (x^2 - 3) (x^2 - \sqrt 5)
- A. Suppose \int\limits_0^1 f(t) dt = 3. Find \int\limits_1^{1.5} f(3 - 2t) dt. B. Let f be twice differentiable with f(0) = 6, f(1) = 5, and f'(1) = 2. Find \int\limits_0^1 xf''(x) dx.
- Let f be a differentiable function such that f(3) = 15, f(6) = 3, f'(3) = -8, and f'(6) = -2. The function g is differentiable and g(x) = f^(-1)(x) for all x. What is the value g'(3)? (A) -1/2 (B) 1/8 (C) 1/6 (D) 1/3
- Suppose g is the inverse function of a differentiable function g and G(x) = \dfrac{1}{ g(x)} . If f(3) = 2 \enspace and \enspace f'(3) = \frac{1 }{9} , find G''(2)
- Suppose g is the inverse function of a differentiable function f and G(x) = \frac{1}{g(x)} . Given f(2)=3, f'(2) =1/4 , find G'(3)
- Suppose f ? 1 is the inverse function of a differentiable function f and f ( 4 ) = 5 , f ? ( 4 ) = 2 3 . Find ( f ? 1 ) ? ( 5 ) .