Product rule with a power function and cosine

Let $f(x) = x^4 \cos(x)$ . Find $f'(x)$ .

Hint

This is a product of two functions, $x^4$ and $\cos(x)$ . Apply the product rule: $\frac{d}{dx}[u\cdot v] = u'v + uv'$ , using $u = x^4$ and $v = \cos(x)$ .

Step-by-step solution

Identify the two factors: $u(x) = x^4$ and $v(x) = \cos(x)$ . Their derivatives are $u'(x) = 4x^3$ and $v'(x) = -\sin(x)$ .

Apply the product rule $f'(x) = u'v + uv'$ : $f'(x) = 4x^3\cos(x) + x^4(-\sin(x)) = 4x^3\cos(x) - x^4\sin(x)$

Product rule with a power function and cosine

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