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Calculadora de Derivada Parcial

Calcular derivadas parciais passo a passo

Esta calculadora online calculará a derivada parcial da função, com as etapas mostradas. Você pode especificar qualquer ordem de integração.

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)`.

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Solution

Your input: find $$$\frac{\partial}{\partial k}\left(\frac{k x}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}\right)$$$

Apply the constant multiple rule $$$\frac{\partial}{\partial k} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial k} \left(f \right)$$$ with $$$c=\frac{x}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}$$$ and $$$f=k$$$:

$${\color{red}{\frac{\partial}{\partial k}\left(\frac{k x}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}\right)}}={\color{red}{\frac{x}{m x \left(x - 1\right) + n \left(x - 1\right) + 1} \frac{\partial}{\partial k}\left(k\right)}}$$

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

$$\frac{x {\color{red}{\frac{\partial}{\partial k}\left(k\right)}}}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}=\frac{x {\color{red}{1}}}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}$$

Thus, $$$\frac{\partial}{\partial k}\left(\frac{k x}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}\right)=\frac{x}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}$$$

Answer: $$$\frac{\partial}{\partial k}\left(\frac{k x}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}\right)=\frac{x}{m x \left(x - 1\right) + n \left(x - 1\right) + 1}$$$