Originally Posted by

**Sudharaka** Dear Amberosia32,

For the first and second problems you should use the chain rule. That is if y and z are functons of x, $\displaystyle \frac{dy}{dx}=\frac{dy}{dz}\times\frac{dz}{dx}$.

eg: $\displaystyle \frac{d(sin^{2}x)}{dx}=\frac{d(sinx)^2}{d(sinx)}\t imes{\frac{(sinx)}{dx}}=2sinxcosx=sin2x$

Similar method could be used to differentiate $\displaystyle cos^{2}x$ and $\displaystyle e^{5x}$

For a curve in two dimention(including a independent and dependent variable.) $\displaystyle \frac{dy}{dx}$ gives the tangent of the curve.

Therefore you could differentiate the equation, and afterwards subject $\displaystyle \frac{dy}{dx}$,

$\displaystyle x^{4} + 6xy +y^{2} = 29$

$\displaystyle 4x^{3}+6x\frac{dy}{dx}+6y+2y\frac{dy}{dx}=0$

Subject $\displaystyle \frac{dy}{dx}$ and you would get,

$\displaystyle \frac{dy}{dx}=-\frac{(2x^{3}+3y)}{3x+y}$

Here I have used the Leibniz's theorem in addition to the chain rule,

That is if v and u are two functions of x, $\displaystyle \frac{d(vu)}{dx}=v\frac{du}{dx}+u\frac{dv}{dx}$

Hope these things will help you to understand differentiation. But if you have any questions please don't hesitate to reply me.