Part 5/6:
Example 4: Product and Chain Rule Together
Finally, we conclude with a function that requires both the Product Rule and the Chain Rule:
( f(x) = (2x + 1)^5 (x^3 - x + 1)^4 )
Deriving this function involves the following steps:
Applying the Product Rule, where we differentiate each part separately.
The derivative of ( (2x + 1)^5 ) is ( 10(2x + 1)^4 \cdot 2 ).
The derivative of ( (x^3 - x + 1)^4 ) employs the Chain Rule, yielding ( 4(x^3 - x + 1)^3(3x^2 - 1) ).
The final form is a combination of both derivatives structured appropriately, which is optimal for evaluation in examinations.