SHORT NOTES: Differentiation

Derivative of a function:
For $y = f\left( x \right)$ ,

$\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}=\frac{dy}{dx}=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=f'(x)$

Derivative of a sum and difference:

If $y=u(x)\pm v(x)$ ,

$\frac{dy}{dx}=\frac{du(x)}{dx}\pm\frac{dv(x)}{dx}$

Differentiation of term:

If $y=k.f(x)$

$\frac{dy}{dx}=k.\frac{df(x)}{dx}$

Product rule:

If $y=u(x).v(x)$ ,

$\frac{dy}{dx}=v(x).\frac{du(x)}{dx}+u(x).\frac{dv(x)}{dx}$

Quotient rule:

If $y=\frac{u(x)}{v(x)}$

$\frac{dy}{dx}=\frac{v(x).\frac{du(x)}{dx}-u(x).\frac{dv(x)}{dx}}{{v(x)}^2}$

Chain rule:

If $y=u\left \{ v(x) \right \}$

$\frac{dy}{dx}=\frac{du\left \{ v(x) \right \}}{d\left \{ v(x) \right \}}\times \frac{dv(x)}{dx}$

Derivatives of some standard functions:

Note: co means negative(-)(Like derivative of Cos,Cot,Cosec results in negative )

Example of chain rule:

Find the derivative of $y=\cos \ (lnx)^2$

Solution:

Let $v=ln \ x$ , $u=(ln \ x)^2$ $\Rightarrow u=v^2,y=\cos u$

So by chain rule,

$\frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dv}.\frac{dv}{dx}$

$\frac{dy}{du}=\frac{d}{du}cos \ u=-\sin u$

$\frac{du}{dv}=\frac{d}{dv}v^2=2v$

$\frac{dv}{dx}=\frac{d}{dv}ln \ x=\frac{1}{x}$

$\therefore \frac{dy}{dx}=-2sin \ (ln \ x)^2 .\ \frac{ln \ x}{x}$

ELECTRICAL ENGINEERING
ENGINEERING MATHEMATICS	Electric Circuits and Fields
Signals and Systems	Electrical Machines
Power Systems	Control Systems
Electrical and Electronic Measurements	Power Electronics and Drives
Analog Electronics	Digital Electronics