# A complicated derivative using chain rule

1. Mar 26, 2012

1. The problem statement, all variables and given/known data

I have a function z, and I need to find the derivative dz/dt "using the chain rule without substitution"

2. Relevant equations

$$z = x^{2}y^{3} + e^{y}\cos x$$
$$x = \log(t^{2})$$
$$y = \sin(4t)$$

3. The attempt at a solution

$$\frac{\mathrm{d} z}{\mathrm{d} t} = 2xy^{3}+3x^{2}y^{2}+e^{y}\cos x-e^{y}\sin x$$
$$\frac{\mathrm{d} z}{\mathrm{d} t} = 2\log(t^{2})\sin(4t)^{3}+3(\log(t^{2}))^{2}(\sin(4t))^{2}+e^{\sin(4t)}(\cos (\log(t^{2}))-\sin (\log(t^{2})))$$
$$\frac{\mathrm{d} z}{\mathrm{d} t} = 4\log(t)\sin(4t)^{3}+3(2\log(t))^{2}(sin(4t))^{2}+e^{sin(4t)}(\cos (2\log(t))-sin (2\log(t)))$$

I'm a bit stuck at this point, I feel like I should be able to do something with the log^2 and the e^sin . (cos(log) - sin(log))?

Last edited: Mar 26, 2012
2. Mar 27, 2012

### clamtrox

That's not right.. Now you have x=x(t), y=y(t), z=z(t) so you'd get
$$\frac{dz}{dt} = 3 x^2 y^2 \frac{dy}{dt} + ...$$
etc. This is the chain rule, and it's the most important differentiation rule, so you should definitely read on it further :-)

3. Mar 29, 2012