Q. \[ \frac{d}{dx}(x+y) \]

Q: What is \frac{d}{dx}(x+y) if y is constant?

\frac{d}{dx}(x+y)=\frac{d}{dx}x+\frac{d}{dx}y=1+0=1 .

Q: What is \frac{d}{dx}(x+y) if y=y(x)?

\frac{d}{dx}(x+y(x))=1+\frac{dy}{dx} .

Q: Is \frac{d}{dx}(x+y) equal to \frac{dx}{dx}+\frac{dy}{dx} ?

Yes, since differentiation is linear: \frac{d}{dx}(x+y)=\frac{d}{dx}x+\frac{d}{dx}y=1+\frac{dy}{dx} .

Q: How do you differentiate x+y using the sum rule?

Apply the sum rule: \frac{d}{dx}(u+v)=\frac{du}{dx}+\frac{dv}{dx} . Here u=x, v=y.

Q: What is the derivative of y+x with respect to x?

Same result by commutativity of addition: \frac{d}{dx}(y+x)=\frac{dy}{dx}+1 .

Q: If y=5, what is \frac{d}{dx}(x+y) ?

Then \frac{dy}{dx}=0 , so \frac{d}{dx}(x+5)=1 .

Answer

To find the derivative of \(x+y\), use the fact that derivatives are taken term by term.

\[
\frac{d}{dx}(x+y)=\frac{d}{dx}(x)+\frac{d}{dx}(y)=1+0=1
\]

Final result: \(1\)

Detailed Explanation

Let’s find the derivative of \(x+y\) with respect to the variable (which is typically \(x\)).

Step 1: Identify the function

The function is

\[
f(x)=x+y
\]
Here, \(y\) is typically treated as a constant with respect to \(x\) (unless the problem states that \(y\) depends on \(x\)).

Step 2: Differentiate term by term

By the sum rule, the derivative of a sum is the sum of the derivatives:

\[
\frac{d}{dx}(x+y)=\frac{d}{dx}(x)+\frac{d}{dx}(y)
\]

Step 3: Differentiate \(x\)

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

Step 4: Differentiate \(y\)

If \(y\) is a constant with respect to \(x\), then

\[
\frac{d}{dx}(y)=0
\]

Step 5: Add the results

\[
\frac{d}{dx}(x+y)=1+0=1
\]

Final Answer

\[
\frac{d}{dx}(x+y)=1
\]

See full solution

Struggling with derivatives? Try AI homework help now!

AI Homework Helper

Calculus FAQ

What is \( \frac{d}{dx}(x+y) \) if \(y\) is constant?

\( \frac{d}{dx}(x+y)=\frac{d}{dx}x+\frac{d}{dx}y=1+0=1 \).

What is \( \frac{d}{dx}(x+y) \) if \(y=y(x)\)?

\( \frac{d}{dx}(x+y(x))=1+\frac{dy}{dx} \).

Is \( \frac{d}{dx}(x+y) \) equal to \( \frac{dx}{dx}+\frac{dy}{dx} \)?

Yes, since differentiation is linear: \( \frac{d}{dx}(x+y)=\frac{d}{dx}x+\frac{d}{dx}y=1+\frac{dy}{dx} \).

How do you differentiate \(x+y\) using the sum rule?

Apply the sum rule: \( \frac{d}{dx}(u+v)=\frac{du}{dx}+\frac{dv}{dx} \). Here \(u=x\), \(v=y\).

What is the derivative of \(y+x\) with respect to \(x\)?

Same result by commutativity of addition: \( \frac{d}{dx}(y+x)=\frac{dy}{dx}+1 \).

If \(y=5\), what is \( \frac{d}{dx}(x+y) \)?

Then \( \frac{dy}{dx}=0 \), so \( \frac{d}{dx}(x+5)=1 \).

Use these AI tools to practice.
Get help with derivative x+y.

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

AI Math Solver Geometry AI Trig Solver