Questions: Find and simplify the derivative of the following function. h(x) = (2x^9 + 2x)(8x^3 + 4x^2 + 4)

Transcript text: Find and simplify the derivative of the following function. \[ h(x)=\left(2 x^{9}+2 x\right)\left(8 x^{3}+4 x^{2}+4\right) \]

Solution

Solution Steps

To find and simplify the derivative of the given function, we will use the product rule. The product rule states that the derivative of two functions multiplied together is the derivative of the first function times the second function plus the first function times the derivative of the second function. We will apply this rule to the given function and then simplify the result.

Step 1: Identify the Function and Apply the Product Rule

Given the function: \[ h(x) = (2x^9 + 2x)(8x^3 + 4x^2 + 4) \]

We apply the product rule, which states: \[ (fg)' = f'g + fg' \]

Let \( f(x) = 2x^9 + 2x \) and \( g(x) = 8x^3 + 4x^2 + 4 \).

Step 2: Differentiate Each Function

Calculate the derivatives: \[ f'(x) = \frac{d}{dx}(2x^9 + 2x) = 18x^8 + 2 \] \[ g'(x) = \frac{d}{dx}(8x^3 + 4x^2 + 4) = 24x^2 + 8x \]

Step 3: Apply the Product Rule

Substitute into the product rule: \[ h'(x) = (18x^8 + 2)(8x^3 + 4x^2 + 4) + (2x^9 + 2x)(24x^2 + 8x) \]

Step 4: Simplify the Expression

Simplify the expression: \[ h'(x) = 192x^{11} + 88x^{10} + 72x^8 + 64x^3 + 24x^2 + 8 \]