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Chapter 3: Problem 10

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\cos x^{5}$$

Short Answer

Expert verified
Answer: The derivative of \(y = \cos{(x^5)}\) with respect to \(x\) is \(\frac{dy}{dx} = -5x^4 \sin{(x^5)}\).

Step by step solution

01

Identify the functions

We have the functions: - \(g(u) = \cos{u}\), - \(u = f(x) = x^5\).
02

Find the derivatives of g and f

We need to find the derivatives of g and f: 1. \(\frac{dg}{du} = -\sin{u}\), 2. \(\frac{df}{dx} = 5x^4\).
03

Apply the Chain Rule

Using the Chain Rule, we have: \(\frac{dy}{dx} = \frac{dg}{du} \times \frac{df}{dx}\).
04

Plug in the values of derivatives

Now we plug in the values of \(\frac{dg}{du}\) and \(\frac{df}{dx}\) that we found earlier: \(\frac{dy}{dx} = (-\sin{u}) \times (5x^4)\).
05

Replace u with f(x)

Since \(u = f(x) = x^5\), we can replace u in the previous expression: \(\frac{dy}{dx} = (-\sin{x^5}) \times (5x^4)\).
06

Simplify the expression

The final expression for the derivative is: $$\frac{dy}{dx} = -5x^4 \sin{(x^5)}.$$

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Most popular questions from this chapter

The following limits equal the derivative of a function \(f\) at a point a. a. Find one possible \(f\) and \(a\) b. Evaluate the limit. $$\lim _{x \rightarrow \pi / 4} \frac{\cot x-1}{x-\frac{\pi}{4}}$$

Special Product Rule In general, the derivative of a product is not the product of the derivatives. Find nonconstant functions \(f\) and \(g\) such that the derivative of \(f g\) equals \(f^{\prime} g^{\prime}\).

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Suppose \(y=L(x)=a x+b\) (with \(a \neq 0\) ) is the equation of the line tangent to the graph of a one-to-one function \(f\) at \(\left(x_{0}, y_{0}\right) .\) Also, suppose that \(y=M(x)=c x+d\) is the equation of the line tangent to the graph of \(f^{-1}\) at \(\left(y_{0}, x_{0}\right)\) a. Express \(a\) and \(b\) in terms of \(x_{0}\) and \(y_{0}\) b. Express \(c\) in terms of \(a,\) and \(d\) in terms of \(a, x_{0},\) and \(y_{0}\) c. Prove that \(L^{-1}(x)=M(x)\)
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