Why is the product rule used when finding the partial derivative for...

[joey_junior]

Why is the product rule used when finding the partial derivative (with respect to a variable) when e is in the function? I noticed it's usually treated as a separate function within a function.

For example, I'm trying to find z'_t(x,y) when z = e^{x^2} + y²e^xy with x = 2t + 3s and y = t²s³

I've found the correct partial derivatives with respect to x, as well as both for t, but apparently the partial derivative of z with respect to y is 2ye^xy + xy²e^xy. They've clearly broke y²e^xy down into two separate functions; (y²) and (e^xy) and differentiated using product rule, but I don't know why, as my textbook does a poor job of showing the proof. It just seems so arbitrary. So basically I'm just wondering why this is the case, and what I should look for when trying to spot when to use the product rule when finding the partial derivatives of variables in similar functions. Thank you.

 

[daon2]

Why is the product rule used when finding the partial derivative (with respect to a variable) when e is in the function? I noticed it's usually treated as a separate function within a function.

For example, I'm trying to find z'_t(x,y) when z = e^{x^2} + y²e^xy with x = 2t + 3s and y = t²s³

I've found the correct partial derivatives with respect to x, as well as both for t, but apparently the partial derivative of z with respect to y is 2ye^xy + xy²e^xy. They've clearly broke y²e^xy down into two separate functions; (y²) and (e^xy) and differentiated using product rule, but I don't know why, as my textbook does a poor job of showing the proof. It just seems so arbitrary. So basically I'm just wondering why this is the case, and what I should look for when trying to spot when to use the product rule when finding the partial derivatives of variables in similar functions. Thank you.

\(\displaystyle \underbrace{y^2}_{f(y)}\cdot \underbrace{e^{xy}}_{g(y)}\) is a product of two functions of \(\displaystyle y\), so the product rule is needed here in finding \(\displaystyle z_y\).

To find \(\displaystyle z_x\), the same part of \(\displaystyle z\) is only a constant (with respect to x) times a function of \(\displaystyle x\). \(\displaystyle \underbrace{y^2}_{\text{constant}}\cdot \underbrace{e^{xy}}_{h(x)}\).

 

[joey_junior]

\(\displaystyle \underbrace{y^2}_{f(y)}\cdot \underbrace{e^{xy}}_{g(y)}\) is a product of two functions of \(\displaystyle y\), so the product rule is needed here in finding \(\displaystyle z_y\).

To find \(\displaystyle z_x\), the same part of \(\displaystyle z\) is only a constant (with respect to x) times a function of \(\displaystyle x\). \(\displaystyle \underbrace{y^2}_{\text{constant}}\cdot \underbrace{e^{xy}}_{h(x)}\).

Oh, I think I sort of understand it now. So basically, if a variable occurs multiple times in a function, as either a constant or exponent of another constant, you have to break it down into smaller functions and use the product rule when finding the partial derivatives with respect to that variable? And I guess this means it's not exclusive to exponential functions as well, correct?

Thank you so much.