Product and Quotient Rules
The Rules
Product Rule:
\(\(\frac{d}{dx}uv=v \frac{du}{dx}+u \frac{dv}{dx}\)\)
\(\((uv)^{'}=u(v)^{'}+(u)^{'}v\)\)
Quotient Rule:
\(\(\frac{d}{dx} \frac{u}{v}= \frac{v \frac{du}{dx}-u \frac{dv}{dx}}{v^2}\)\)
Differentiating tan(x)
- \(\tan(x)=\frac{\sin(x)}{\cos(x)}\)
- Applying product rule:
- \(\frac{\cos x\times \cos x-\sin x\times-\sin x}{\cos^2x}\)
- \(\frac{\cos^2x+\sin^2x}{\cos^2x}\)
- \(\frac{1}{\cos^2x}=sec^2x\)
Exercise 9D Q1
\(x(1+3x)^{5}\)
\(u=x\)
\(u^{'}=1\)
\(v=(1+3x)^5\)
\(v^{'}=15(1+3x)^{4}\)
\(15x(1+3x)^{4}+(1+3x)^5\)
\((1+3x)^4[15x+1+3x]\)
\((1+3x)^4\times(1+18x)\)
Example Question
-
\[\frac{dy}{dx}\frac{\sin4x}{x^3}=\frac{(4\cos4x\times x^3)-(3x^2\times \sin4x)}{x^6}\]
-
\[\frac{4\cos4x}{x^3}-\frac{3\sin 4x}{x^4}\]