Questions: Find the derivative of the following function by first expanding the expression. f(x)=(5x+3)(6x^2+1)

Find the derivative of the following function by first expanding the expression. f(x)=(5x+3)(6x^2+1)
Transcript text: Find the derivative of the following function by first expanding the expression. \[ f(x)=(5 x+3)\left(6 x^{2}+1\right) \]
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Solution

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Solution Steps

To find the derivative of the function \( f(x) = (5x + 3)(6x^2 + 1) \), first expand the expression by distributing the terms. Once expanded, differentiate each term with respect to \( x \).

Step 1: Expand the Expression

To find the derivative of the function \( f(x) = (5x + 3)(6x^2 + 1) \), we first expand the expression. Distributing the terms, we have: \[ f(x) = 30x^3 + 18x^2 + 5x + 3 \]

Step 2: Differentiate the Expanded Function

Next, we differentiate the expanded function term by term with respect to \( x \): \[ f'(x) = \frac{d}{dx}(30x^3) + \frac{d}{dx}(18x^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(3) \] Calculating each derivative, we get: \[ f'(x) = 90x^2 + 36x + 5 \]

Final Answer

\(\boxed{f'(x) = 90x^2 + 36x + 5}\)

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