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):

`(d(f\cdot g))/(dx)(x)=(df)/(dx)\cdot g+f\cdot(dg)/(dx)`

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

`(dh)/(dx)(x)=(d(f\cdot g))/(dx)(x)=(d(x^3))/(dx)(5-3x)^4+x^3(d((5-3x)^4))/(dx)`
`\begin{gathered} (d(x^3))/(dx)=3x^2 \\ (d((5-3x)^4))/(dx)=4\cdot(5-3x)^3\cdot(d(5-3x))/(dx) \\ (d(5-3x))/(dx)=(d(5))/(dx)-(d(3x))/(dx)=0-3=-3 \end{gathered}`
`\begin{gathered} (dh)/(dx)=3x^2(5-3x)^4+x^3\cdot4\cdot(5-3x)^3\cdot(-3) \\ (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} (dh)/(dx)=3x^2(5-3x)^3\lbrack(5-3x)-4x\rbrack \\ (dh)/(dx)=3x^2(5-3x)^3(5-7x) \end{gathered}`