Find the value of partial z / partial x at the point (1, 1, 1) if the equation 8 x y + z^3 x - 4...


Find the value of {eq}\displaystyle \dfrac {\partial z}{\partial x} {/eq} at the point {eq}(1,\ 1,\ 1) {/eq} if the equation {eq}\displaystyle 8 x y + z^3 x - 4 y z = 5 {/eq} defines {eq}z {/eq} as a function of the two independent variables {eq}x {/eq} and {eq}y {/eq} and the partial derivative exists.

Partial Differentiation:

Partial differentiation of a function of two or more variables with respect to a particular variable is obtained by differentiating the fuction with only one variable and keeping the remaining variables constant at the same time. It represents the rate of change of the function with respect to only variables. Some formulas of differentiation that are generally utilized to differentiate a function are listed below:

$$\begin{align} \\ &\hspace{1cm} \dfrac{\text{d}c}{\text{d}x} =0 & & \left[\text{ Where } c\ \text{is constant value } \right]\\[0.3cm] &\hspace{1cm} \dfrac{\text{d}}{\text{d}x} x^n =nx^{n-1} & & \left[\text{ This is power rule of differentiation } \right]\\[0.3cm] &\hspace{1cm}\displaystyle \dfrac{\text{d}}{\text{d}x} \left[ uv \right]=u \frac{\text{d}v}{dx}+v\frac{\text{d}u}{dx} && \left[\text{ This is product rule of differentiation } \right]\\[0.3cm] &\hspace{1cm} \frac{\text{d}}{\text{d}x}f(g(x)) =f'(g(x)) g'(x) && \left[\text{ This is chain rule of differentiation } \right]\\[0.3cm] \end{align}\\ $$

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  • {eq}\displaystyle 8 x y + z^3 x - 4 y z = 5\; , P(1,\ 1,\ 1) {/eq}


{eq}\displaystyle \dfrac {\partial z}{\partial x}=\;? {/eq}


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Higher-Order Partial Derivatives Definition & Examples


Chapter 14 / Lesson 2

Learn what partial derivatives and higher order partial derivatives are. Find out how to solve higher and second order partial derivatives with examples.

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