Question

Use the Chain Rule to find the derivative. $$ y=\sec \frac{1}{x} $$

   Use the Chain Rule to find the derivative.
$$
y=\sec \frac{1}{x}
$$

Calculus for AP

Calculus for AP

Jon Rogawski & Colin… 3rd Edition

Chapter 3, Problem 37

Instant Answer

Solved by Expert Linh Vu

10/17/2020

Step 1

In this case, the inner function is $u=\frac{1}{x}$ and the outer function is $y=\sec(u)$. Show more…

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Use the Chain Rule to find the derivative. $$ y=\sec \frac{1}{x} $$

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Key Concepts

Derivative of Reciprocal Functions

Functions such as 1/x are considered reciprocal functions and can be rewritten as x^(-1). Differentiating them involves applying the power rule, resulting in an expression like -x^(-2). This technique is commonly used as the inner function differentiation step in a chain rule setup.

Composite Functions

A composite function is formed when one function is applied to the result of another function. Differentiating composite functions requires identifying the inner and outer functions and then applying the chain rule, which is essential for handling nested or layered expressions.

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. It states that if you have a function composed of an outer function and an inner function, the derivative of the composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

Derivative of Trigonometric Functions

Knowing the derivative of trigonometric functions is crucial when differentiating functions like secant. Specifically, the derivative of secant is sec(x)tan(x), which is used as the outer function's derivative when applying the chain rule to composite functions involving secant.

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