Questions: g(x)=(5x^2+2x+1)e^x find derivative

Solution Steps

To find the derivative of the function \( g(x) = (5x^2 + 2x + 1)e^x \), we will use the product rule of differentiation. The product rule states that if you have a function \( h(x) = f(x) \cdot g(x) \), then the derivative \( h'(x) \) is given by \( h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \). Here, \( f(x) = 5x^2 + 2x + 1 \) and \( g(x) = e^x \).

1. Differentiate \( f(x) = 5x^2 + 2x + 1 \) to get \( f'(x) \).
2. Differentiate \( g(x) = e^x \) to get \( g'(x) \).
3. Apply the product rule: \( g'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \).

We start with the function \( f(x) = 5x^2 + 2x + 1 \). The derivative of \( f(x) \) is calculated as follows: \[ f'(x) = \frac{d}{dx}(5x^2 + 2x + 1) = 10x + 2 \]

Next, we differentiate the function \( g(x) = e^x \). The derivative is: \[ g'(x) = \frac{d}{dx}(e^x) = e^x \]

Using the product rule, we find the derivative of \( g(x) = (5x^2 + 2x + 1)e^x \): \[ g'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \] Substituting the derivatives we found: \[ g'(x) = (10x + 2)e^x + (5x^2 + 2x + 1)e^x \]

We can factor out \( e^x \) from the expression: \[ g'(x) = \left((10x + 2) + (5x^2 + 2x + 1)\right)e^x \] Combining the terms inside the parentheses: \[ g'(x) = (5x^2 + 12x + 3)e^x \]

Final Answer