Question

If $f(x)=(2x+1)^4$ , then the 4th derivative of $f(x)$ at x=0 is what ?

Answer 1

simply appply the chain rule over and over again
f'(g(x))=f'(g(x))g'(x)

first derivitive would be f'(x)=4(2x+1)³(2) or 8(2x+1)³
2nd derivitive would be 24(2x+1)²(2) or 48(2x+1)²
3rd derivitive would be 96(2x+1)2 or 192(2x+1)
4th derivitive would be 192(1)(2) or 384
at x=0, the derivitive is 384

Answer 2

The 4th derivative of $f(x)$ at $x=0$ is 384.

Given:

The given function is:

$f(x)=(2x+1)^4$

To find:

The 4th derivative of $f(x)$ at $x=0$.

Explanation:

Differentiate the given function with respect to $x$.

$f'(x)=4(2x+1)^3(2x+1)'$

$f'(x)=4(2x+1)^3(2)$

$f'(x)=8(2x+1)^3$

Differentiate the above function with respect to $x$.

$f''(x)=8\cdot 3(2x+1)^2(2x+1)'$

$f''(x)=24(2x+1)^2(2)$

$f''(x)=48(2x+1)^2$

Differentiate the above function with respect to $x$.

$f'''(x)=48\cdot 2(2x+1)^1(2x+1)'$

$f'''(x)=96(2x+1)^1(2)$

$f'''(x)=192(2x+1)$

Differentiate the above function with respect to $x$.

$f''''(x)=192(2+0)$

$f''''(x)=384$

At $x=0$, the value of $f''''(0)=384$. Therefore, the required value is 384.

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