# Need help taking partial derivatives

by paul2001
Tags: derivatives, partial
 P: 2 Hello everyone, I am trying to take partial derivatives of the following equation and I am having difficulties. The partial derivatives are of Q w.r.t D, ΔP, ρ, and w. Any help would be much appreciated. Thank you. Paul Attached Thumbnails
Math
Emeritus
Thanks
PF Gold
P: 39,549
 Quote by paul2001 Hello everyone, I am trying to take partial derivatives of the following equation and I am having difficulties. The partial derivatives are of Q w.r.t D, ΔP, ρ, and w. Any help would be much appreciated. Thank you. Paul
The function is
$$Q= 5.9863 CD^2\left[\frac{\Delta P}{\rho(1-\beta^4)}\right]^{0.5}$$

As a function of D, that is just D2 and the derivitive of that is 2D.
$$Q_P= 2(5.9863) CD\left[\frac{\Delta P}{\rho(1-\beta^4)}\right]^{0.5}$$

As a function of $\Delta P$, that is just $(\Delta P)^{0.5}$ and the derivative of that is $0.5(\Delta P)^{-0.5}$
$$Q_{\Delta P}= 0.5 (5.9863) CD^2\left[\frac{1}{\Delta P\rho(1-\beta^4)}\right]^{0.5}$$

As a function of $\rho$ it is $1/\rho^{0.5}= \rho^{-0.5}$ and the derivative of that is $-0.5\rho^{-1.5}$
$$Q_{\rho}= -0.5(5.9863) CD^2\left[\frac{\Delta P}{\rho^3(1-\beta^4)}\right]^{0.5}$$

I do not see any "w" in the formula so the derivative with respect to "w" would be 0!

There is a "C" that you did not mention.
$$Q_C= 5.9863 D^2\left[\frac{\Delta P}{\rho(1-\beta^4)}\right]^{0.5}$$

There is also a "$\beta$ in the formula. That's a little more complicated because the function involves $(1- \beta^4)^{-0.5}$ and the derivative of that, by the chain rule, is $-0.5(1- \beta^4)^{-1.5}(-4\beta^3)= 2(1- \beta^4)^{-1.5}\beta^3$
$$Q_{\beta}= 2(5.9863) CD^2\beta^3\left[\frac{\Delta P}{\rho(1-\beta^{12}}\right]^{0.5}$$

Since this has nothing directly to do with differential equations, I am moving it to "Calculus and Analysis".
 P: 75 Just pretend that whatever variable you're not taking the partial derivative with respect to is a constant. For example if you're taking it with respect to B, just treat all of the other variables like constants, and differentiate it the way you would: $$\sqrt{1 - B^4}$$ using the chain rule since everything else is just a constant multiple that isn't going to be impacted by integration.
 P: 2 Need help taking partial derivatives I have attached the same equation with the appropriate substitutions. The problem I am having is that the equation is iterative so I cannot simply take the partial derivatives without solving for Q. Is there a special approach I can use for this type of derivation? Thank you again. Paul Attached Thumbnails
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,549 Well, you don't have to solve for Q, you can use "implict differentiation".
 P: 75 Yeah, just put dQ/dwhateva on both sides and then take it like both sides were the expression.

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