Question

Evaluate (d^100/(dx^95 dy^2 dx^3))(ye^x(x)+cos(x)).

Answer 1

\[\frac{\partial^{100}}{\partial x^{95}\partial y^2\partial x^3}\left(yxe^{x}+\cos x\right)\]
If you're familiar with higher-order partial derivatives, you'd know that
\[\frac{\partial^2f}{\partial x\partial y}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial y}\right)\]
The point is that you should find the derivatives starting from the right side; in other words, find \(\dfrac{\partial^3 }{\partial x^3}(yxe^{x}+\cos x)\), then \(\dfrac{\partial ^2}{\partial y^2}(yxe^x+\cos x)\).

I stop there for reasons you'll see soon enough.

Answer 2

What do you mean? How is the function different?

And it has to be partial derivatives. The way it's written with normal d's doesn't make sense. We're dealing with a multivariable function here.

Answer 3

@Loser66, Implicit differentiation doesn't make sense in this context. You can only do that if you're given an equation that cannot be explicitly written with one variable as a function of another.

We're not given an equation, just some expression containing x's and y's. So you must assume \(f(x,y)=yxe^x+\cos x\), and hence the partial derivatives.