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Chapter 3: Problem 11

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=e^{5 x-7}$$

Short Answer

Expert verified
Answer: The derivative of the function \(y = e^{5x - 7}\) with respect to \(x\) is \(\frac{dy}{dx} = 5e^{5x-7}\).

Step by step solution

01

Find the Inner Function (u)

The given function is \(y = e^{5x-7}\). Identify the inner function as \(u = 5x - 7\).
02

Differentiate the Inner Function (u)

Differentiate the inner function \(u = 5x - 7\) with respect to \(x\) to get its derivative, \(\frac{du}{dx}\): $$\frac{du}{dx} = 5$$
03

Replace the Inner Function with u in the Outer Function

Replace the inner function in the given function \(y= e^{5x-7}\) with \(u\) to find the outer function: $$y = e^u$$
04

Differentiate the Outer Function (y)

Differentiate the outer function \(y=e^u\) with respect to \(u\) to get its derivative, \(\frac{dy}{du}\): $$\frac{dy}{du} = e^u$$
05

Apply the Chain Rule

Using the Chain Rule formula \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\), substitute \(\frac{du}{dx} = 5\) and \(\frac{dy}{du} = e^u\) from steps 2 and 4: $$\frac{dy}{dx} = e^u \cdot 5$$
06

Replace u with the Original Inner Function

In the expression \(\frac{dy}{dx} = e^u \cdot 5\), replace \(u\) with the original inner function \(u=5x-7\): $$\frac{dy}{dx} = 5e^{5x-7}$$ The derivative of the given function \(y = e^{5x - 7}\) with respect to \(x\) is: $$\frac{dy}{dx} = 5e^{5x-7}$$

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Most popular questions from this chapter

Cobb-Douglas production function The output of an economic system \(Q,\) subject to two inputs, such as labor \(L\) and capital \(K\) is often modeled by the Cobb- Douglas production function \(Q=c L^{a} K^{b} .\) When \(a+b=1,\) the case is called constant returns to scale. Suppose \(Q=1280, a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=40\) a. Find the rate of change of capital with respect to labor, \(d K / d L\) b. Evaluate the derivative in part (a) with \(L=8\) and \(K=64\)

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Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\) . $$f(x)=-x^{2}+8 ;(7,1)$$

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=x^{2 / 3}, \text { for } x>0$$

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{f(x)}{g(x)}\right]\right|_{x=2}$$
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