Question

Find the second derivative of the following function: F(x) = (5-2x)^4

Answer 1

The given equation is

`(5-2x)^4`

to find the second derivative.

for that let us find the first derivative,

use the formula,

`(d)/(dx)(x^n)=nx^(n-1)`

apply the chian rule,

`\begin{gathered} =4(5-2x)(d)/(dx)(5-2x) \\ =4\mleft(5-2x\mright)^3\mleft(-2\mright) \\ =-8\mleft(5-2x\mright)^3 \end{gathered}`

Now, let us find the second derivative.

`(d)/(dx)(-8(5-2x)^3)`

take out the constant,

`-8(d)/(dx)((5-2x)^3)`

use the formula,

`\begin{gathered} (d)/(dx)(x^n)=nx^(n-1) \\ \end{gathered}`

apply the chain rule,

`\begin{gathered} =3(5-2x)^2(d)/(dx)(5-2x) \\ =-8\cdot\: 3(5-2x)^2(-2) \\ =48\mleft(5-2x\mright)^2 \end{gathered}`

the answer is

`48(5-2x)^2`