Evaluate the derivatives of the following functions. 1. f ( x ) = x \operatorname { sin } ^ { - 1...
Question:
Evaluate the derivatives of the following functions.
1. {eq}f ( x ) = x \operatorname { sin } ^ { - 1 } x {/eq}
2. {eq}g ( z ) = \operatorname { tan } ^ { - 1 } ( 1 / z ) {/eq}
3. {eq}f ( t ) = ( \operatorname { cos } ^ { - 1 } t ) ^ { 2 } {/eq}
Derivative of inverse Trigonometric:
Suppose a function, {eq}f(x) {/eq}, where one or more factors or terms contain the inverse of trigonometric.
Then, to find the derivative of {eq}f(x) {/eq}, we may apply some differentiation rules, such as, Chain rule, Product rule or Quotient rule and apply the differentiation formula of inverse trigonometric,
{eq}\dfrac{d}{dx}\sin^{-1}x = \dfrac{1}{\sqrt{1-x^2}} \\ \dfrac{d}{dx}\cos^{-1}x = \dfrac{-1}{\sqrt{1-x^2}} \\ \dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1+x^2}. {/eq}
Answer and Explanation: 1
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View this answer1. The derivative of {eq}f ( x ) = x \sin ^ { - 1 } x {/eq} is evaluated by applying Product rule, {eq}(uv)' = u'v +uv'. {/eq}
Then,
{eq}\begin{a...
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Chapter 8 / Lesson 3Explore several different examples of trigonometric functions, their equations, and graphs. Learn how to calculate the derivatives of trigonometric functions.