Question

Find $d y / d x$. $$ y=\cot ^{-1}(\sqrt{x}) $$

   Find $d y / d x$.
$$
y=\cot ^{-1}(\sqrt{x})
$$

Calculus Early Transcendentals

Calculus Early Transcendentals

Howard Anton, Irl… 9th Edition

Chapter 3, Problem 51

Instant Answer

Solved by Expert Zachary Mitchell

08/26/2020

Step 1

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here, our outer function is $\cot ^{-1}(u)$ and our inner function is $\sqrt{x}$. Show more…

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Find $d y / d x$. $$ y=\cot ^{-1}(\sqrt{x}) $$

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Key Concepts

Differentiation of Radical Functions

Differentiating radical functions, such as square root functions, is a core concept in calculus. The derivative of a square root function involves applying the power rule by rewriting the expression as a fractional exponent. Mastery of this concept is important when such functions appear as the inner part of a composite function, ensuring the overall differentiation is carried out correctly.

Chain Rule

The chain rule is a fundamental differentiation tool used to compute the derivative of composite functions. When a function is nested within another function, as in the case where an inverse trigonometric function is applied to a radical function, the chain rule allows the derivative of the outer function to be multiplied by the derivative of the inner function. This method is essential for accurately differentiating complex expressions.

Inverse Trigonometric Differentiation

This concept involves finding derivatives of functions defined as the inverse of standard trigonometric functions. Each inverse trigonometric function, such as the inverse cotangent, has a well-established derivative formula. Understanding these formulas is crucial for differentiating expressions where the inverse trigonometric function is applied to another function, allowing one to compute the derivative systematically.

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