Q. \(\frac{d}{dx}\left(xe^x\right)\)

Q: Compute \frac{d}{dx}(xe^x) . \

Use the product rule: \frac{d}{dx}(xe^x)=x\frac{d}{dx}(e^x)+e^x\frac{d}{dx}(x)=xe^x+e^x=e^x(x+1) .

Q: Why do we use the product rule for xe^x ? \

Because it’s a product of two functions: x and e^x. Apply \frac{d}{dx}(uv)=u'v+uv' with u=x, v=e^x.

Q: Find \frac{d}{dx}(x)e^x + x\frac{d}{dx}(e^x) . \

Differentiate each part: \frac{d}{dx}(x)=1 and \frac{d}{dx}(e^x)=e^x . So 1\cdot e^x + x\cdot e^x = e^x(1+x)=e^x(x+1) .

Q: Compute the second derivative of xe^x . \

First derivative: (xe^x)'=e^x(x+1) . Then (e^x(x+1))' = e^x(x+1)+e^x = e^x(x+2) .

Q: Solve (xe^x)'=e^x(x+1) by expanding. \

From product rule: (xe^x)' = x(e^x)' + (x)'e^x = xe^x + e^x . Factor e^x: e^x(x+1) .

Answer

To find \\(\frac{d}{dx}(xe^x)\\), use the product rule. Let \\(u=x\\) and \\(v=e^x\\). Then \\(u’=1\\) and \\(v’=e^x\\).

\[
\frac{d}{dx}(xe^x)=u’v+uv’=1\cdot e^x+x\cdot e^x=e^x+xe^x=e^x(1+x).
\]

Final result: \(\frac{d}{dx}(xe^x)=e^x(1+x)\).

Detailed Explanation

We want to find the derivative of the function \(f(x)=x e^x\).

Step 1: Identify the function type.

The function is a product of two functions:

\(u(x)=x\)
\(v(x)=e^x\)

Step 2: Use the Product Rule.

The Product Rule says:

\[
\frac{d}{dx}\bigl(u(x)v(x)\bigr)=u'(x)v(x)+u(x)v'(x).
\]

Step 3: Compute each derivative.

Differentiate \(u(x)=x\):

\[
u'(x)=\frac{d}{dx}(x)=1.
\]

Differentiate \(v(x)=e^x\):

\[
v'(x)=\frac{d}{dx}(e^x)=e^x.
\]

Step 4: Substitute into the Product Rule.

Now apply:

\[
\frac{d}{dx}\bigl(x e^x\bigr)=u'(x)v(x)+u(x)v'(x).
\]

Substitute \(u'(x)=1\), \(v(x)=e^x\), \(u(x)=x\), and \(v'(x)=e^x\):

\[
\frac{d}{dx}\bigl(x e^x\bigr)=1\cdot e^x + x\cdot e^x.
\]

Step 5: Simplify.

Both terms contain \(e^x\), so factor it out:

\[
\frac{d}{dx}\bigl(x e^x\bigr)=e^x(1+x).
\]

Final Answer:

\[
\frac{d}{dx}\bigl(x e^x\bigr)=e^x(x+1).
\]

See full solution

Need help with derivatives? Try our AI homework tools!

AI Homework Helper

Calculus FAQ

Compute \( \frac{d}{dx}(xe^x) \). \

Use the product rule: \( \frac{d}{dx}(xe^x)=x\frac{d}{dx}(e^x)+e^x\frac{d}{dx}(x)=xe^x+e^x=e^x(x+1) \).

Why do we use the product rule for \( xe^x \)? \

Because it’s a product of two functions: \(x\) and \(e^x\). Apply \( \frac{d}{dx}(uv)=u'v+uv' \) with \(u=x\), \(v=e^x\).

Find \( \frac{d}{dx}(x)e^x + x\frac{d}{dx}(e^x) \). \

Differentiate each part: \( \frac{d}{dx}(x)=1 \) and \( \frac{d}{dx}(e^x)=e^x \). So \(1\cdot e^x + x\cdot e^x = e^x(1+x)=e^x(x+1) \).

Compute the second derivative of \( xe^x \). \

First derivative: \( (xe^x)'=e^x(x+1) \). Then \( (e^x(x+1))' = e^x(x+1)+e^x = e^x(x+2) \).

Solve \( (xe^x)'=e^x(x+1) \) by expanding. \

From product rule: \( (xe^x)' = x(e^x)' + (x)'e^x = xe^x + e^x \). Factor \(e^x\): \( e^x(x+1) \).

Compute the derivative using factoring first: \( xe^x = x\cdot e^x \). \

Factoring doesn’t change the need for product rule. Differentiate \(x\) and \(e^x\): \( (xe^x)' = x\cdot e^x + 1\cdot e^x = e^x(x+1) \).

Use math AI tools for clear steps.
Check derivatives of xe^x easily.

298,376+ active customers
Math, Geometry, Trigonometry, etc.

AI Math Solver Geometry AI Trig Solver