Find the derivative of each function. (a) f(x)=cos√(x) (b) f(x)=√(cosx) (c) f(x)=cos(1)/(2) x

Name: Problem 20, Chapter 2: Calculus: Early Transcendental Functions
Uploaded: 2022-10-07T11:33:44-08:00
Channel: Amelia Hardy
Description: Find the derivative of each function. (a) f(x)=cos√(x) (b) f(x)=√(cosx) (c) f(x)=cos(1)/(2) x

step 1 Here I want to find the derivative of each function. So I'm going to start by rewriting my first

Find the derivative of each function. (a) f(x)=cos√(x) (b)...

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. It states that if a function y = f(g(x)) is composed of an outer function f and an inner function g, then its derivative is given by multiplying the derivative of f evaluated at g(x) by the derivative of g(x). This concept is crucial when dealing with nested functions, such as a trigonometric function of a square root or a square root of a trigonometric function.

Differentiation of Trigonometric Functions

Differentiation of trigonometric functions involves applying known differentiation formulas, such as the derivative of cosine being -sine. When these functions are part of a composite setup, their usual differentiation rules must be combined with the chain rule. This approach is used to differentiate functions where the trigonometric function is applied to another function of x.

Differentiation of Radical Functions

Differentiating a radical function, particularly a square root, involves recognizing that a square root can be expressed as a fractional exponent (x^(1/2)) and then applying the power rule. When a square root function is part of a composite function, the derivative is found by using the chain rule in combination with the power rule.

Constant Multiple Rule

The constant multiple rule in differentiation states that if a function is multiplied by a constant, the derivative of the product is that constant multiplied by the derivative of the function. This rule simplifies the differentiation process when dealing with functions that include constant factors, whether as multipliers in the function argument or in the function itself.

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