# Differentiation rule, Theorem Examples with Solutions

Now we will prove the important formula or rules which are used to determine the derivative of different functions efficiently.

You can prove differentiation in easy way by using following pattren. Step 1: Let function.........eq(1) Step 2: adding \delta y and \delta x in y and x component respectively .......eq(2) Step 3: Subtracting eq(2)-eq(1) Step 4: Dividing \delta x on both sides. Step 5: applying \underset{\delta x\rightarrow0}{lim} on both sides.

Step 1       :Let function

Let y=cf(x).......(1)

Step 2:                                   adding \delta y and \delta x in y and x component respectively

y+\delta y=cf\left(x+\delta x\right)........(2)

Step 3:                                                   eq(2)-eq(1)

y+\delta y-y=cf\left(x+\delta x\right)-cf\left(x\right)

\delta y=c\left[f\left(x+\delta x\right)-f\left(x\right)\right]

Step 4:                                      dividing \delta x on both sides

\frac{\delta y}{\delta x}=\frac{c\left[f\left(x+\delta x\right)-cf\left(x\right)\right]}{\delta x}

Step 5:                                      Taking \underset{\delta x\rightarrow0}{lim}

\lim_{\delta x\rightarrow0}\frac{\delta y}{\delta x}=\lim_{\delta x\rightarrow0}\left[c\frac{f\left(x+\delta x\right)-cf\left(x\right)}{\delta x}\right]

\lim_{\delta x\rightarrow0}\frac{\delta y}{\delta x}=\underset{\delta x\rightarrow0}{\left(c\right)\lim}\left[\frac{f\left(x+\delta x\right)-cf\left(x\right)}{\delta x}\right] c is constant and constant factor can be taken out from a limit sign

\frac{dy}{dx}=cf'\left(x\right)

\boxed{\frac{dy}{dx}=c\frac d{dx}f\left(x\right)}

Example 1