Respuesta :

The given equation is

[tex](5-2x)^4[/tex]

to find the second derivative.

for that let us find the first derivative,

use the formula,

[tex]\frac{d}{dx}(x^n)=nx^{n-1}[/tex]

apply the chian rule,

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

Now, let us find the second derivative.

[tex]\frac{d}{dx}(-8(5-2x)^3)[/tex]

take out the constant,

[tex]-8\frac{d}{dx}((5-2x)^3)[/tex]

use the formula,

[tex]\begin{gathered} \frac{d}{dx}(x^n)=nx^{n-1} \\ \end{gathered}[/tex]

apply the chain rule,

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

the answer is

[tex]48(5-2x)^2[/tex]