Question

find the derivative of h(x)=x^3 (5 -3x)^4

Answer 1

We can solve this with the derivative of f(x) multiplied by g(x):

$\frac{d(f\cdot g)}{dx}(x)=\frac{df}{dx}\cdot g+f\cdot\frac{dg}{dx}$

In this case we can choose f(x)=x^3 and g(x)=(5-3x)^4, so:

$\frac{dh}{dx}(x)=\frac{d(f\cdot g)}{dx}(x)=\frac{d(x^3)}{dx}(5-3x)^4+x^3\frac{d((5-3x)^4)}{dx}$$\begin{gathered} \frac{d(x^3)}{dx}=3x^2 \\ \frac{d((5-3x)^4)}{dx}=4\cdot(5-3x)^3\cdot\frac{d(5-3x)}{dx} \\ \frac{d(5-3x)}{dx}=\frac{d(5)}{dx}-\frac{d(3x)}{dx}=0-3=-3 \end{gathered}$$\begin{gathered} \frac{dh}{dx}=3x^2(5-3x)^4+x^3\cdot4\cdot(5-3x)^3\cdot(-3) \\ \frac{dh}{dx}=3x^2(5-3x)^4-12x^3(5-3x)^3 \end{gathered}$

We can factor by 3x^2(5-3x)^3 to simplify the formula:

$\begin{gathered} \frac{dh}{dx}=3x^2(5-3x)^3\lbrack(5-3x)-4x\rbrack \\ \frac{dh}{dx}=3x^2(5-3x)^3(5-7x) \end{gathered}$