Differentiation wrt to a parameter

Question

If I have an equation

$$2(a+b\cos(x))^2\dot y$$

Where $\dot y$ is the first derivative of $y(u)$ wrt to $u$

Where $a,b$ are constants and $x,y$ are functions of a parameter $u$. How would one differentiate this equation wrt to $u$?

Answer 1

$$2(a+b\cos x)^2y'=2a^2+4aby'\cos x + 2b^2y'\cos^2x $$

Differentiating, we should get:

$$\frac{\mathrm d }{\mathrm du}(2a^2+4aby'\cos x \ + 2b^2y'\cos^2x)\\ =0+4ab(y''\cos x-y'x'\sin x)+2b^2\left(y''\cos^2x+ y'(x'\sin x)^2 \right)$$

where $y'$ is the first derivate of $y$ w.r.t $u$.

In Newton notation: $$4ab(\ddot y\cos x-\dot y\dot x\sin x)+2b^2\left(\ddot y\cos^2x+ \dot y(\dot x\sin x)^2 \right)$$

Answer 2

You would differentiate as normal using the product ,quotient and chain rules as you would normally do .

$I = 2(a+b\cos(x))^2\dot y$

differentiate wrt u

$\frac{dI}{du} = 2(a+b\cos(x))^2\cdot\ddot y+ 4(a+b\cos(x))\cdot (-b\sin(x))\cdot\dot x $

where the derivatives are in Newton notation wrt $u $.