Question

3-34: Differentiate the function.

19. \(f\left( x \right) = {x^3}\left( {x + 3} \right)\)

Short answer

The derivative of the function is \(f'\left( x \right) = 4{x^3} + 9{x^2}\).

Step-by-step solution

 Differentiation Rule

When \(c\) is a constant and \(f\) is a differentiable function, then;

\(\frac{d}{{dx}}\left( {cf\left( x \right)} \right) = c\frac{d}{{dx}}f\left( x \right)\)

\(\frac{d}{{dx}}\left( x \right) = 1\)

The derivative of the constant function is;

\(\frac{d}{{dx}}\left( c \right) = 0\)

When \(f\) and \(g\) are both differentiable, then;

\(\begin{array}{l}\frac{d}{{dx}}\left( {f\left( x \right) + g\left( x \right)} \right) &=& \frac{d}{{dx}}f\left( x \right) + \frac{d}{{dx}}g\left( x \right)\\\frac{d}{{dx}}\left( {f\left( x \right) - g\left( x \right)} \right) &=& \frac{d}{{dx}}f\left( x \right) - \frac{d}{{dx}}g\left( x \right)\end{array}\)

Differentiate the function

Rewrite the function as shown below:

\(\begin{aligned}f\left( x \right) &= {x^3}\left( {x + 3} \right)\\ &= {x^3} \cdot x + 3{x^3}\\ &= {x^4} + 3{x^3}\end{aligned}\)

Differentiate the function as shown below:

\(\begin{aligned}f'\left( x \right) &= \frac{d}{{dx}}\left( {{x^4} + 3{x^3}} \right)\\ &= \frac{d}{{dx}}\left( {{x^4}} \right) + 3\frac{d}{{dx}}\left( {{x^3}} \right)\\ &= 4{x^{4 - 1}} + 3\left( 3 \right){x^{3 - 1}}\\ &= 4{x^3} + 9{x^2}\end{aligned}\)

Thus, the derivative of the function is \(f'\left( x \right) = 4{x^3} + 9{x^2}\).