Question - Derivative of Product of Functions
Solution:
The given functions are $$ f(x) = x^3 + 1 $$ and $$ g(x) = x^2 $$. We need to find the derivative of the product of these functions, $$ f(x) \cdot g(x) $$, at the point $$ x = -1 $$.To solve this, we first find the product of the functions:\[ f(x) \cdot g(x) = (x^3 + 1)(x^2) = x^5 + x^2 \]Now we take the derivative of the product:\[ \frac{d}{dx}(f(x) \cdot g(x)) = \frac{d}{dx}(x^5 + x^2) \]Using the power rule for derivatives $$ \frac{d}{dx}(x^n) = nx^{n-1} $$, the derivatives of each term are:\[ \frac{d}{dx}(x^5) = 5x^4 \]\[ \frac{d}{dx}(x^2) = 2x \]Combine them:\[ \frac{d}{dx}(f(x) \cdot g(x)) = 5x^4 + 2x \]Finally, we evaluate this derivative at $$ x = -1 $$:\[ \frac{d}{dx}(f(x) \cdot g(x))|_{x = -1} = 5(-1)^4 + 2(-1) = 5 - 2 = 3 \]Therefore, the derivative of the function $$ f(x) \cdot g(x) $$ at the point $$ x = -1 $$ is 3.
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