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Chapter 3: Problem 7

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=(3 x+7)^{10}$$

Short Answer

Expert verified
Question: Given the function y=(3x+7)^{10}, find its derivative with respect to x. Answer: The derivative of the given function with respect to x is: \(\frac{dy}{dx} = 30(3x+7)^{9}\)

Step by step solution

01

Identify the inner and outer functions

We are given the function $$y=(3x+7)^{10}.$$ The inner function is \(u = 3x + 7\) and the outer function is \(y = u^{10}\).
02

Differentiate the outer function without considering the inner function

Apply the power rule for derivatives to the outer function \(y = u^{10}\): $$\frac{dy}{du} = 10u^{9}$$ Remember that we have not considered the inner function yet.
03

Differentiate the inner function

Now we will find the derivative of the inner function, \(u = 3x + 7\): $$\frac{du}{dx} = 3$$
04

Apply the Chain Rule

The Chain Rule states: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ We already found both derivatives in the previous steps, therefore: $$\frac{dy}{dx} = (10u^{9}) \cdot (3)$$
05

Substitute the inner function back into the derivative

Recall that \(u = 3x + 7\). Now we substitute this back into our derivative: $$\frac{dy}{dx} = 10(3x+7)^{9} \cdot 3$$
06

Simplify the derivative

Finally, we can simplify the expression: $$\frac{dy}{dx} = 30(3x+7)^{9}$$ The derivative of the given function with respect to x is: $$\frac{dy}{dx} = 30(3x+7)^{9}$$

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