Calculadora de derivadas parciales
Calcular derivadas parciales paso a paso
Esta calculadora en línea calculará la derivada parcial de la función, con los pasos que se muestran. Puede especificar cualquier orden de integración.
Solution
Your input: find $$$\frac{\partial}{\partial y}\left(81 x^{2} + y^{2}\right)$$$
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}{\frac{\partial}{\partial y}\left(81 x^{2} + y^{2}\right)}}={\color{red}{\left(\frac{\partial}{\partial y}\left(81 x^{2}\right) + \frac{\partial}{\partial y}\left(y^{2}\right)\right)}}$$The derivative of a constant is 0:
$${\color{red}{\frac{\partial}{\partial y}\left(81 x^{2}\right)}} + \frac{\partial}{\partial y}\left(y^{2}\right)={\color{red}{\left(0\right)}} + \frac{\partial}{\partial y}\left(y^{2}\right)$$Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=2$$$:
$${\color{red}{\frac{\partial}{\partial y}\left(y^{2}\right)}}={\color{red}{\left(2 y^{-1 + 2}\right)}}=2 y$$Thus, $$$\frac{\partial}{\partial y}\left(81 x^{2} + y^{2}\right)=2 y$$$
Answer: $$$\frac{\partial}{\partial y}\left(81 x^{2} + y^{2}\right)=2 y$$$