Калькулятор частинних похідних

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

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

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

The derivative of a constant is 0:

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

Apply the constant multiple rule $\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$ with $c=3$ and $f=x$:

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

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

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

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

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