The given function is y = 5x2 e4x+12.

Question

ScanSolve AI Tutor · Accepted Answer

Okay, Nurse, let's find the derivative of this function.

The given function is $y = 5x^2 e^{4x+12}$ .
We need to find $\frac{dy}{dx}$ . This requires the product rule and the chain rule.

Identify the two functions in the product.
Let $u = 5x^2$ and $v = e^{4x+12}$ .
The product rule states that if $y = uv$ , then $\frac{dy}{dx} = u'v + uv'$ .
Differentiate $u$ with respect to $x$ .
$u = 5x^2$ $u' = \frac{d}{dx}(5x^2) = 5 \cdot 2x = 10x$
Differentiate $v$ with respect to $x$ using the chain rule.
$v = e^{4x+12}$
The derivative of $e^{f(x)}$ is $e^{f(x)} \cdot f'(x)$ .
Here, $f(x) = 4x+12$ .
$f'(x) = \frac{d}{dx}(4x+12) = 4$
So,
$v' = \frac{d}{dx}(e^{4x+12}) = e^{4x+12} \cdot 4 = 4e^{4x+12}$
Apply the product rule formula.
Substitute $u$ , $u'$ , $v$ , and $v'$ into the product rule formula: $\frac{dy}{dx} = u'v + uv'$ .
$\frac{dy}{dx} = (10x)(e^{4x+12}) + (5x^2)(4e^{4x+12})$
Simplify the expression.
$\frac{dy}{dx} = 10x e^{4x+12} + 20x^2 e^{4x+12}$
Factor out the common terms, which are $10x e^{4x+12}$ .
$\frac{dy}{dx} = 10x e^{4x+12} (1 + 2x)$
The final answer is $\textbf{10x}e^{\textbf{4x+12}}\textbf{(1 + 2x)}$ .

Answer