Question

would implicit differentiation work on the problem below. derivative with 3 variables

Answer 1

\[4x ^{3} - \ln y +\sqrt[3]{zx} = \sin yx + 2\ln z\]

Answer 2

this is single or multivariable calc?

Answer 3

multi

Answer 4

^ btw thats made up , just wondering if you would just follow all regular derivative rules then get some answer

Answer 5

in multivariable calculus you have to decide which derivative you are differentiating with respect to, as well as if one variable is a function of the other. Is z a function of x for example.

Answer 6

You can partially differentiate implicitly wrt a certain variable. Assuming all variables are independent and differentiating implicitly wrt to x would give\[4x ^{3} - \ln y +\sqrt[3]{zx} = \sin yx + 2\ln z\]\[12x^2=y\cos yx\]so not much interesting happening there

Answer 7

But if z is a function of, say, x and y\[z=f(x,y)\]and we differentiate partially wrt x, we get\[12x^2+\frac13(zx)^{-2/3}\left(x\frac{\partial z}{\partial x}+z\right)=y\cos yx+\frac2z\frac{\partial z}{\partial x}\]That can be solved for \(\large\frac{\partial z}{\partial x}\) just like a implicit differentiation for single-variable calc.

Answer 8

@Goten77 do you follow?

Answer 9

http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx#ImplicitDiffSingle

Answer 10

i havent taken a calculus class that high of level yet but it makes sense