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I know this is basic and I've read up on partial derivatives but to double check and make sure I have it correct. I have three problems where I need to calculate the partial derivative with respect to X. Z is another variable. C is constant but unknown:

  1. Y = Z*X*5
  2. Y = XC + ZC + $15x^{2}$
  3. y = $X^{1/2}$ + Z

My answers are as follows:

  1. 5Z (is it 5Z or just 5?)
  2. Z + 15
  3. Z(or is it $.5X^{-1/2}$)

Any and all help would be greatly appreciated. If my answers are wrong, please provide the correct one so I can work backwards to solve the problem. Thank you.

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  • $\begingroup$ There is a diference between $x$ and $X$? $\endgroup$ – Rafael Wagner Mar 30 '17 at 17:39
  • $\begingroup$ no there is not sorry. $\endgroup$ – Marley Chatthews Mar 30 '17 at 18:23
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For #2, consider the function $f(x)=xc$, where $c$ is a constant. Surely you agree that $f'(x)=c$, not $f'(x)=0$. Your calculation for 2 is quite off.

Assuming $x$ and $X$ are the same variable, the answer to 2 should be $\frac{\partial}{\partial X} Y = C+30X$.

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    $\begingroup$ $\dfrac d{dC}$ would be $X+Z$. It looks like OP did a mix of the two. In fact, one could say that OP... partially did each (sorry). $\endgroup$ – tilper Mar 30 '17 at 15:18
  • $\begingroup$ So is the answer for #3 $.5x^{-.5}$ + z? Thanks for the help. $\endgroup$ – Marley Chatthews Mar 30 '17 at 15:52
  • $\begingroup$ @MarleyChatthews No, it would not. Perhaps it would help if you take a look at Wikipedia's description of a partial derivative: $$ $$ "In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary)." $$ $$ Therefore, the $z$ term should be regarded as a constant. Now, what is the derivative of a constant? $\endgroup$ – projectilemotion Mar 30 '17 at 17:12
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    $\begingroup$ I see....so would just be $.5x^{-.5}$? $\endgroup$ – Marley Chatthews Mar 30 '17 at 18:33
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Well, you have the functions: $$y=5xz \tag{1}$$ $$y=cx+cz+15x^2 \tag{2}$$ $$y=x^{1/2}+z \tag{3}$$ Perhaps it would help to solve the problems if you take a look at Wikipedia's description of a partial derivative:

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).


Your solution to the first one is correct. Since we are taking partial derivatives with respect to $x$, we consider $5z$ as a constant. Therefore: $$\frac{\partial}{\partial x}(5xz)=5z\cdot \frac{\partial}{\partial x}(x)=5z$$


The second one is incorrect. We consider $cz$ as a constant since it does not contain $x$. Recall that the derivative of a constant is $0$. $$\frac{\partial}{\partial x}(cx+cz+15x^2)=\frac{\partial}{\partial x}(cx)+\color{red}{\frac{\partial}{\partial x}(cz)}+\frac{\partial}{\partial x}(15x^2)$$ $$=c\cdot \frac{\partial}{\partial x}(x)+15\cdot \frac{\partial}{\partial x}(x^2)=c+30x$$


Do you think you can do the third one?

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    $\begingroup$ is it $.5x^{-.5}$ $\endgroup$ – Marley Chatthews Mar 30 '17 at 18:35
  • $\begingroup$ @MarleyChatthews Yes, that's right. $\endgroup$ – projectilemotion Mar 30 '17 at 18:37

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