# Math Help - Partial Derivative

1. ## Partial Derivative

How would I do this partial derivative?

$\frac {d^2z}{dydx} ...........z= \frac{(x+y)^{y+1}}{y+1} +sinh^3(y)+2$

Thanks.

2. Hello,

Firstly, derivate in function of x.

The derivative of $\frac{(x+y)^{y+1}}{y+1}$ is $(x+y)^y$ (do you know why ?)

The derivative of $sinh^3(y)+2$ is 0 because there is no x in it, so it's like a constant if the variable is x.

Now remains $(x+y)^y$ to derivate in function of y.

$(x+y)^y=e^{y \ln(x+y)}$

The derivative for $e^{f(x)}$ is $f'(x)e^{f(x)}$

Here, $f(y)=y \ln(x+y)$. So, by using the product rule, we have : $f'(y)=\frac{y}{x+y}+\ln(x+y)$

Hence $\frac{d^2z}{dydx}=e^{y \ln(x+y)}(\frac{y}{x+y}+\ln(x+y))$ $=(x+y)^y \frac{y}{x+y}+(x+y)^y \ln(x+y)=y(x+y)^{y-1}+(x+y)^y \ln(x+y)$