Example Question - product of functions
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Derivative of Product of Functions
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.
Multiplication of Two Functions
The question asks for the multiplication of two functions f(x) and g(x), where: f(x) = 2x + 5 g(x) = x^2 - 5x To find the product (f * g)(x), you multiply each term in f(x) by each term in g(x): (f * g)(x) = (2x + 5)(x^2 - 5x) Now expand the product: = 2x * x^2 + 2x * (-5x) + 5 * x^2 + 5 * (-5x) = 2x^3 - 10x^2 + 5x^2 - 25x = 2x^3 - (10x^2 - 5x^2) - 25x = 2x^3 - 5x^2 - 25x Therefore, the result is: (f * g)(x) = 2x^3 - 5x^2 - 25x This polynomial is already in its simplest form.
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