# Find {eq}f'(x) {/eq} if {eq}f(x) = x^3 (x^2 + 3)^3 {/eq}

## Question:

Find {eq}f'(x) {/eq} if {eq}f(x) = x^3 (x^2 + 3)^3 {/eq}

## Product Rule of Derivative:

The given function {eq}f(x) {/eq} represents a multiplication of two functions. If we speculated these two functions are {eq}g(x) \, and \, h(x) {/eq} then we can write:

{eq}g(x) = x^3 \, or \, (x^2 +3)^3 {/eq}

And:

{eq}h(x) = x^3 \, or \, (x^2 +3)^3 {/eq}

Now:

We are imploring to locate {eq}f'(x) {/eq} and we realize that {eq}f'(x) {/eq} is computed by the Product Rule of Derivative.

The Rule is clarified below:

Product Rule of Derivative:

{eq}\frac{d}{dx} (f(x)g(x)) = f(x) \frac{d}{dx} (g(x)) + g(x) \frac{d}{dx} (f(x)) {/eq}

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The function is given by:

{eq}f(x) = x^3 (x^2 + 3)^3 {/eq}

Derivative of {eq}f(x) {/eq} with respect to {eq}x {/eq} is:

{eq}\begin{align*} f^...

Derivatives: The Formal Definition

from

Chapter 7 / Lesson 5
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The derivative in calculus is the rate of change of a function. In this lesson, explore this definition in greater depth and learn how to write derivatives.