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Chapter 3: Problem 8

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\left(5 x^{2}+11 x\right)^{20}$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = (5x^2 + 11x)^{20}\) with respect to \(x\). Answer: \(\frac{dy}{dx} = 20(5x^2 + 11x)^{19} \cdot (10x + 11)\)

Step by step solution

01

Identify the outer and intermediate functions

In this case, the outer function is \(y(u) = u^{20}\) and the intermediate function is \(u(x) = 5x^2 + 11x\).
02

Find the derivatives of the outer and intermediate functions

First, we will find the derivative of the outer function with respect to the intermediate function, \(\frac{dy}{du}\). Using the power rule, we get: $$\frac{dy}{du} = 20u^{19}$$ Then, we will find the derivative of the intermediate function with respect to the given variable \(x\), \(\frac{du}{dx}\). Using the power rule and sum rule, we get: $$\frac{du}{dx} = 10x + 11$$
03

Apply the chain rule

Now, according to the chain rule, we can find the derivative of the composite function \(y\) with respect to \(x\), \(\frac{dy}{dx}\), by multiplying \(\frac{dy}{du}\) and \(\frac{du}{dx}\). To do this, substitute the value of \(u\) back into the derivative of the outer function: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = (20u^{19})\cdot(10x + 11) = 20(5x^2 + 11x)^{19} \cdot (10x + 11)$$ So, the derivative of the function \(y = (5x^2 + 11x)^{20}\) with respect to \(x\) is: $$\frac{dy}{dx} = 20(5x^2 + 11x)^{19} \cdot (10x + 11)$$

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