# Homework Help: Partial differential coefficient

1. Jan 5, 2017

### jonjacson

1. The problem statement, all variables and given/known data

The equation is z= e (x*y), the interesting thing is y is function of x too, y = ψ(x)

Calculate the partial derivative respect to x, and the total derivative.

2. Relevant equations

Total differential:

dz= ∂z/∂x dx + ∂z/∂y dy

3. The attempt at a solution

Well, according to the book:

∂z/∂x= y * exy

But I don't agree with this result, because if y is also function of x this term will be different.

For example if y=x3, z would be = ex4, and the partial derivative is:

∂z/∂x = 4 x3 * e x4

and this is not the same as:

y * e xy = x3ex4

What do you think?

I just want to check if the book is right, this topic (multivariate differential calculus) is so important that I want to understand it correctly.

2. Jan 5, 2017

### Orodruin

Staff Emeritus
The book is right. When they talk about partial derivative they talk about the derivative considering $e^{yx}$ as a function of $x$ and $y$ as independent parameters. What you are computing is the total derivative.

3. Jan 5, 2017

### Ray Vickson

If $z = e^{x\, \psi(x)}$ then there really is no such thing as a "partial derivative" of $z$; there is only an "ordinary" (1-variable) derivative $dz/dx$.
However, that being said, we do have that $z'(x)$ is expressed in terms of the partial derivatives of the function $e^{xy}$ evaluated at $y = \psi(x)$ (and, of course, the derivative $d \psi(x)/dx$ is involved as well).

To fix your difficulties when $y = x^3$ you need to use the full force of the complete chain rule:
$$\frac{d}{dx} \left( \left. e^{x y} \right|_{y = x^3} \right) = \left. \frac{\partial e^{x y}}{\partial x} \right|_{y = x^3} + \left. \frac{ \partial e^{xy}}{\partial y} \right|_{y = x^3} \: \cdot \frac{d \, x^3}{dx}$$