# Partial Derivative Calculator

## Calculate partial derivatives step by step

This online calculator will calculate the partial derivative of the function, with steps shown. You can specify any order of integration.

Enter a function:

Enter the order of integration:

Hint: type x^2,y to calculate (partial^3 f)/(partial x^2 partial y), or enter x,y^2,x to find (partial^4 f)/(partial x partial y^2 partial x).

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

### Solution

Your input: find $\frac{\partial}{\partial y}\left(3 x + 4 y\right)$

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}{\frac{\partial}{\partial y}\left(3 x + 4 y\right)}}={\color{red}{\left(\frac{\partial}{\partial y}\left(3 x\right) + \frac{\partial}{\partial y}\left(4 y\right)\right)}}$$

Apply the constant multiple rule $\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$ with $c=4$ and $f=y$:

$${\color{red}{\frac{\partial}{\partial y}\left(4 y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)={\color{red}{\left(4 \frac{\partial}{\partial y}\left(y\right)\right)}} + \frac{\partial}{\partial y}\left(3 x\right)$$

Apply the power rule $\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$ with $n=1$, in other words $\frac{\partial}{\partial y} \left(y \right)=1$:

$$4 {\color{red}{\frac{\partial}{\partial y}\left(y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)=4 {\color{red}{1}} + \frac{\partial}{\partial y}\left(3 x\right)$$

The derivative of a constant is 0:

$$4 + {\color{red}{\frac{\partial}{\partial y}\left(3 x\right)}}=4 + {\color{red}{\left(0\right)}}$$

Thus, $\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$

Answer: $\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$
The text passage contains a mathematical question about finding the partial derivative of a function, and it provides a step-by-step solution to the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Partial Derivative Calculator

## Calculate partial derivatives step by step

This online calculator will calculate the partial derivative of the function, with steps shown. You can specify any order of integration.

Enter a function:

Enter the order of integration:

Hint: type x^2,y to calculate (partial^3 f)/(partial x^2 partial y), or enter x,y^2,x to find (partial^4 f)/(partial x partial y^2 partial x).

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

### Solution

Your input: find $\frac{\partial}{\partial y}\left(3 x + 4 y\right)$

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}{\frac{\partial}{\partial y}\left(3 x + 4 y\right)}}={\color{red}{\left(\frac{\partial}{\partial y}\left(3 x\right) + \frac{\partial}{\partial y}\left(4 y\right)\right)}}$$

Apply the constant multiple rule $\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$ with $c=4$ and $f=y$:

$${\color{red}{\frac{\partial}{\partial y}\left(4 y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)={\color{red}{\left(4 \frac{\partial}{\partial y}\left(y\right)\right)}} + \frac{\partial}{\partial y}\left(3 x\right)$$

Apply the power rule $\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$ with $n=1$, in other words $\frac{\partial}{\partial y} \left(y \right)=1$:

$$4 {\color{red}{\frac{\partial}{\partial y}\left(y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)=4 {\color{red}{1}} + \frac{\partial}{\partial y}\left(3 x\right)$$

The derivative of a constant is 0:

$$4 + {\color{red}{\frac{\partial}{\partial y}\left(3 x\right)}}=4 + {\color{red}{\left(0\right)}}$$

Thus, $\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$

Answer: $\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$
user
The text passage contains a mathematical question about finding the partial derivative of a function, and it provides a step-by-step solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER