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Chapter 3: Problem 40

Use the Chain Rule to find the derivative of the following functions. $$y=(\cos x+2 \sin x)^{8}$$

Short Answer

Expert verified
Answer: The derivative of the function is $$\frac{dy}{dx} = 8(\cos x + 2\sin x)^7(-\sin x + 2\cos x)$$.

Step by step solution

01

Identify the inner and outer functions

Let's identify the inner function ($$u(x)$$) and the outer function ($$v(u)$$) from the given function $$y$$. The inner function is: $$u(x) = \cos x + 2\sin x$$. The outer function is: $$v(u) = u^8$$, where $$u$$ is the output of the inner function.
02

Find the derivative of the inner function

Now, we will find the derivative of the inner function, $$u(x)$$, with respect to $$x$$. $$u'(x) = \frac{d}{dx}(\cos x + 2\sin x) = -\sin x + 2\cos x$$
03

Find the derivative of the outer function

Next, we should find the derivative of the outer function, $$v(u)$$, with respect to $$u$$. $$v'(u) = \frac{d}{du}(u^8) = 8u^7$$
04

Apply the Chain Rule

Now that we have the derivatives of both the inner and the outer functions, $$u'(x)$$ and $$v'(u)$$, we will apply the Chain Rule to find the derivative of the given function $$y$$, where $$y = v(u(x))$$. $$\frac{dy}{dx} = v'(u(x)) \cdot u'(x)$$ Substituting the derivatives we found in Steps 2 and 3: $$\frac{dy}{dx} = [8(\cos x + 2\sin x)^7] \cdot (-\sin x + 2\cos x)$$
05

Simplify the expression

Lastly, let's simplify the expression for the derivative of $$y$$ with respect to $$x$$: $$\frac{dy}{dx} = 8(\cos x + 2\sin x)^7(-\sin x + 2\cos x)$$ Therefore, the derivative of the function $$y=(\cos x+2\sin x)^8$$ is: $$\frac{dy}{dx} = 8(\cos x + 2\sin x)^7(-\sin x + 2\cos x)$$

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