Question

Find the derivative of the function g(x) = (4x² - 5x - 4)e^x.

Answer 1

The derivative of the function
`\( g(x) = (4x^2 - 5x - 4)e^x \) is \( g'(x) = (8x - 5 + 4x^2 - 5x - 4)e^x \).`

Step-by-step explanation:

To find the derivative of the given function,
`\( g(x) = (4x^2 - 5x - 4)e^x \)`, we apply the product rule. The product rule states that if
`\( u(x) \) and \( v(x) \)` are differentiable functions, then the derivative of their product is given by
`\( (u(x)v(x))' = u'(x)v(x) + u(x)v'(x) \).`

In this case, let
`\( u(x) = 4x^2 - 5x - 4 \)` and
`\( v(x) = e^x \)`. We find the derivatives
`\( u'(x) \) and \( v'(x) \)` to apply the product rule. The derivative of
`\( u(x) \) is \( u'(x) = 8x - 5 \)` and the derivative of
`\( v(x) \) is \( v'(x) = e^x \)`.

Now, applying the product rule, we get
`\( g'(x) = u'(x)v(x) + u(x)v'(x) = (8x - 5)e^x + (4x^2 - 5x - 4)e^x \).`

Finally, simplifying the expression, we arrive at the final answer:
`\( g'(x) = (8x - 5 + 4x^2 - 5x - 4)e^x \)`. This is the derivative of the given function with respect to (x).