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[katex]………eq(1)[/katex] Step 2: adding [katex]\delta y[/katex] and [katex]\delta x[/katex] in [katex]y[/katex] and [katex]x[/katex] component respectively [katex]…….eq(2)[/katex] Step 3: Subtracting [katex]eq(2)-eq(1)[/katex] Step 4: Dividing [katex]\delta x[/katex] on both sides. Step 5: applying [katex]\underset{\delta x\rightarrow0}{lim}[/katex] on both sides.
Step 1 :Let function
Let [katex]y=cf(x)…….(1)[/katex]
Step 2: adding [katex]\delta y[/katex] and [katex]\delta x[/katex] in y and x component respectively
[katex]y+\delta y=cf\left(x+\delta x\right)……..(2)[/katex]
Step 3: [katex]eq(2)-eq(1)[/katex]
[katex]y+\delta y-y=cf\left(x+\delta x\right)-cf\left(x\right)[/katex]
[katex]\delta y=c\left[f\left(x+\delta x\right)-f\left(x\right)\right][/katex]
Step 4: dividing [katex]\delta x[/katex] on both sides
[katex]\frac{\delta y}{\delta x}=\frac{c\left[f\left(x+\delta x\right)-cf\left(x\right)\right]}{\delta x}[/katex]
Step 5: Taking [katex]\underset{\delta x\rightarrow0}{lim}[/katex]
[katex]\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][/katex]
[katex]\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][/katex] c is constant and constant factor can be taken out from a limit sign
[katex]\frac{dy}{dx}=cf’\left(x\right)[/katex]
[katex]\boxed{\frac{dy}{dx}=c\frac d{dx}f\left(x\right)}[/katex]
Example 1