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Find $f'(x)$ where $f(x)=3x\cos(x)$ .

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Find derivatives of radical functions

Full solution

Q. Find

f'(x)

where

f(x)=3x\cos(x)

.

Identify components: Identify the components of the function that will require the use of the product rule. $\newline$ The function $f(x) = 3x \cos(x)$ is a product of two functions, $g(x) = 3x$ and $h(x) = \cos(x)$ .
Recall product rule: Recall the product rule for differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. In formula terms, if $f(x) = g(x)h(x)$ , then $f'(x) = g'(x)h(x) + g(x)h'(x)$ .
Differentiate first function: Differentiate the first function $g(x) = 3x$ . $\newline$ The derivative of $g(x)$ with respect to $x$ is $g'(x) = \frac{d}{dx}(3x) = 3$ .
Differentiate second function: Differentiate the second function $h(x) = \cos(x)$ . $\newline$ The derivative of $h(x)$ with respect to $x$ is $h'(x) = \frac{d}{dx}(\cos(x)) = -\sin(x)$ .
Apply product rule: Apply the product rule using the derivatives from steps $3$ and $4$ . $\newline$ Using the product rule, $f'(x) = g'(x)h(x) + g(x)h'(x) = 3 \cdot \cos(x) + 3x \cdot (-\sin(x))$ .
Simplify derivative: Simplify the expression for the derivative. $f'(x) = 3\cos(x) - 3x\sin(x)$ .

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