Differentiating power towers

Question

Given the function $z=x-y+2xe^{y^2}$, how can I differentiate it to find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$. I know how to take the partial derivative of all the terms, apart from the power tower. I can do it if it's only a single layer, but how does it work when the tower is multiple layers high?

I know that for single layers, the derivative can be calculated using the chain rule for expressions such as $e^{7y}$, but if the expression was, for example, $e^{7^y}$, would I then put the whole exponent in front to obtain $7^y \times e^{7^y}$?

Answer 1

Simply ...

$$z=x-y+2xe^{y^2}$$ $$\partial_y z= -1+2xe^ {y^2}2y$$ $$\partial_y z= -1+4xye^ {y^2}$$ The rule is $$\frac {d}{dx}e^{h(x)}=e^{h(x)}.h'(x)$$

Answer 2

Hint: $e^{y^2}$ is just $\exp(f(y))$, where $\exp(x) = e^x$ and $f(y) = y^2$. Similarly $e^{7^y}$ is just $\exp(g(y))$ where $g(y) = 7^y$. In this way, you can reduce any number of "layers" to an ordinary function composition and then apply the chain rule. (But make sure you calculate the derivatives $f'$ and $g'$ correctly.)