Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ y=\left(2 x^{3}+1\right)^{2} $$

Step-by-Step Solution

Answer

\( y' = 12x^2(2x^3 + 1) \)

Step 1: Understand the General Power Rule

General Power Rule also known as the chain rule in Calculus states that the derivative of \( h(g(x)) \) is \( h'(g(x)) \cdot g'(x) \). As such, we need to identify the outer function \( h(x) \) and the inner function \( g(x) \). For the given function \( y = (2x^3 + 1)^2 \), the outer function, \( h(x) \), is \( x^2 \) and the inner function, \( g(x) \), is \( 2x^3 + 1 \).

Step 2: Find the derivative of the outer function and inner function

Using the power rule, the derivative of the outer function \( h'(x) = 2x \) and the derivative of the inner function \( g'(x) = 6x^2 \).

Step 3: Apply the chain rule

The chain rule states that the derivative of a composition of functions is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Therefore \( y' = h'(g(x)) \cdot g'(x) = 2(2x^3 + 1) \cdot 6x^2 \).

Step 4: Simplify the equation

After substitution, simplify the equation to get the final answer. \( y' = 2(2x^3 + 1) \cdot 6x^2 = 12x^2(2x^3 + 1) \).