To find the second derivative f′′(x) of the function f(x)=4(x2+1)3, we first need to express the function in a form that is easier to differentiate. We can rewrite the function as f(x)=((x2+1)3)1/4. Then, we apply the chain rule and power rule to find the first derivative f′(x). After obtaining the first derivative, we differentiate it again to find the second derivative f′′(x).
Step 1: Define the Function
We start with the function defined as:
f(x)=4(x2+1)3=((x2+1)3)41
Step 2: Calculate the First Derivative
Using the chain rule and power rule, we find the first derivative f′(x):
f′(x)=43(x2+1)43−1⋅2x=23x(x2+1)43⋅(x2+1)1
This simplifies to:
f′(x)=2(x2+1)413x
Step 3: Calculate the Second Derivative
Next, we differentiate f′(x) to find the second derivative f′′(x):
f′′(x)=−43⋅(x2+1)453x2+23⋅(x2+1)411
This can be simplified to:
f′′(x)=(x2+1)2(0.75x2+1.5)(x2+1)43
Final Answer
Thus, the second derivative is:
f′′(x)=(x2+1)2(0.75x2+1.5)(x2+1)43