# Derivative of the composition of two differentiable functions

Calculate $$\frac{\mathrm d}{\mathrm dt} f (g(t^2),g(t^4)),$$ where $f$ is a differentiable function of two variables and $g$ is a differentiable function of one variable. Your answer should be expressed in terms of $f, g$ and their derivatives and/or partial derivatives.

I am assuming it is a partial derivatives question. I have never encountered one like this before. Any help would be much appreciated.

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Do you know how to compute more generally $d/dt (f(x(t), y(t))$ ? – user18119 Nov 27 '12 at 14:18
This is more like an apply chain rule question. The partial derivatives part in the question is because $\frac{df(x,y)}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$ (see Wikipedia) -- I hope I didn't miss any $\partial$ while typing all this – user50600 Nov 27 '12 at 14:19

You presumably have a function $f(x,y)$, and somebody has set $x=g(t^2), y=g(t^4).$ Assuming $f$ doesn't depend on $t$ explicitly, $\frac{\mathrm d}{\mathrm dt} f (x,y)=\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}$. Now insert the given values for $x,y$ and use the chain rule.