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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