Find the value of (f ∘g)^' at the given value of x. f(u)=((u-1)/(u+1))^2, u=g(x)=(1)/(x^2)-1,

Name: Problem 38, Chapter 4: Calculus: Graphical, Numerical, Algebraic
Uploaded: 2020-05-06T14:29:05-08:00
Duration: 7 min 40 s
Channel: Robert Daugherty
Description: Find the value of (f ∘g)^' at the given value of x. f(u)=((u-1)/(u+1))^2, u=g(x)=(1)/(x^2)-1, x=-1

step 1 Section 3 .6, problem number 86, so we're dealing with chain rule derivative. So here they give

Question

Find the value of $(f \circ g)^{\prime}$ at the given value of $x$. $$f(u)=\left(\frac{u-1}{u+1}\right)^{2}, \quad u=g(x)=\frac{1}{x^{2}}-1, \quad x=-1$$

   Find the value of $(f \circ g)^{\prime}$ at the given value of $x$.

$$f(u)=\left(\frac{u-1}{u+1}\right)^{2}, \quad u=g(x)=\frac{1}{x^{2}}-1, \quad x=-1$$

Calculus: Graphical, Numerical, Algebraic

Ross L. Finney,… 5th Edition

Chapter 4, Problem 38

Instant Answer

Solved by Expert Robert Daugherty

05/06/2020

Step 1

This means we need to plug in $g(x)$ into the function $f(u)$. So we have: $(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x^2} - 1\right) = \left(\frac{\frac{1}{x^2} - 1 - 1}{\frac{1}{x^2} - 1 + 1}\right)^2$ Now, we need to find the derivative of $(f \circ g)(x)$ Show more…

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Find the value of $(f \circ g)^{\prime}$ at the given value of $x$. $$f(u)=\left(\frac{u-1}{u+1}\right)^{2}, \quad u=g(x)=\frac{1}{x^{2}}-1, \quad x=-1$$

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Key Concepts

Chain Rule

The chain rule is a fundamental differentiation technique used for computing the derivative of a composite function. It states that the derivative of a function formed by the composition of two or more functions is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.

Differentiation of Composite Functions

Differentiating composite functions involves first recognizing the nested structure where one function is applied to the result of another. This requires applying the chain rule to correctly determine the derivative of the overall function with respect to the independent variable.

Differentiation of Rational Functions

Differentiating rational functions, which are defined as the ratio of two polynomial functions, often makes use of the quotient rule. This concept is important for finding derivatives when the function involves ratios, which are common in algebraic expressions.

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