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Chapter 1: Problem 4

Now find the derivative of each of the following functions. \(f(x)=e^{x \cos x}\)

Short Answer

Expert verified
The derivative of the function \(f(x) = e^{x \cos x}\) is \(f'(x) = e^{x \cos x} * (\cos x - x \sin x)\).

Step by step solution

01

Define the Inner Function

Firstly, identify the inner function that's inside the parentheses. Here, the inner function for the equation \(f(x) = e^{x \cos x}\) is \(u = x \cos x\).
02

Differentiate the Inner Function

Let's find the derivative of this inner function, \(u = x \cos x\). Utilising the product rule that states \(f'(x) = u'v + uv'\), where u = x and v = cos x, therefore \(u' = 1\) and \(v' = -\sin x\). Consequently the derivative is: \(u' = 1*\cos x + x*(-\sin x) = \cos x - x \sin x\).
03

Apply the Chain Rule

The chain rule of derivatives states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. The outer function is \(e^u\), with derivative \(e^u\). Applying the chain rule gives \(f'(x) = e^u * u'\). Substitute \(u = x \cos x\) and \(u' = \cos x - x \sin x\) from previous steps. Thus, the derivative of \(f(x) = e^{x \cos x}\) is \(f'(x) = e^{x \cos x} * (\cos x - x \sin x)\).

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