Find the derivative $$ y=x^{2} e^{-3 x}+x $$ a) $$ y \prime=-3 x^{2} e^{-3 x}+2 x e^{-3 x}+1 $$ b) $$ y \prime=2 x e^{-3 x}+x^{2} e^{-3 x}+1 $$ c) $$ y \prime=3 x^{2} e^{-3 x}+2 e^{-3 x} $$ d) $$ u \prime=2 x e^{-3 x}+1 $$

Question

UpStudy AI · Accepted Answer

We are given
$$
y = x^2 e^{-3x} + x.
$$
Differentiate using the product rule for the term $$ x^2 e^{-3x} $$.

1. Let $$ u = x^2 $$ and $$ v = e^{-3x} $$. Then,
   $$
   u' = 2x \quad \text{and} \quad v' = -3e^{-3x}.
   $$
2. Applying the product rule,
   $$
   \frac{d}{dx}\left(x^2 e^{-3x}\right) = u'v + uv' = 2x e^{-3x} + x^2 (-3e^{-3x}) = 2x e^{-3x} - 3x^2 e^{-3x}.
   $$
3. The derivative of $$ x $$ is
   $$
   \frac{d}{dx}(x) = 1.
   $$
4. Combine the results to obtain the derivative $$ y' $$:
   $$
   y' = 2x e^{-3x} - 3x^2 e^{-3x} + 1.
   $$

Thus, the correct answer is

a) $$ y' = -3x^2 e^{-3x} + 2x e^{-3x} + 1 $$.
The derivative is $$ y' = -3x^2 e^{-3x} + 2x e^{-3x} + 1 $$.

Find the derivative

a)
b)
c)
d)

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Calculus Questions

Find the derivative a) b) c) d)