Theorems on Derivatives
1. \(\frac{d}{dx}\{{{f}_{1}}(x)\pm {{f}_{2}}(x)\}=\frac{d}{dx}{{f}_{1}}(x)\pm \frac{d}{dx}{{f}_{2}}(x)\).
2. \(\frac{d}{dx}(kf(x))=k\times \frac{d}{dx}f(x)\) where k is any constant.
3. \(\frac{d}{dx}\{{{f}_{1}}(x)\times {{f}_{2}}(x)\}={{f}_{1}}(x)\frac{d}{dx}{{f}_{2}}(x)+{{f}_{2}}(x)\frac{d}{dx}{{f}_{1}}(x)\).
In general
\(\frac{d}{dx}\{{{f}_{1}}(x).{{f}_{2}}(x).{{f}_{3}}(x)…..\}=\left( \frac{d}{dx}{{f}_{1}}(x) \right)({{f}_{2}}(x).{{f}_{3}}(x))…..+\)
\( \left( \frac{d}{dx}{{f}_{2}}(x) \right)({{f}_{1}}(x).{{f}_{3}}(x)…..)+\left( \frac{d}{dx}{{f}_{3}}(x) \right)({{f}_{1}}(x).{{f}_{2}}(x)…..)+……\),
4. \(\frac{d}{dx}\left\{ \frac{{{f}_{1}}(x)}{{{f}_{2}}(x)} \right\}=\frac{{{f}_{2}}(x)\frac{d}{dx}{{f}_{1}}(x)-{{f}_{1}}(x)\frac{d}{dx}{{f}_{2}}(x)}{{{\{{{f}_{2}}(x)\}}^{2}}}\).
Example: Find \(\frac{dy}{dx}\) for y = x sinx logx
Solution: Given that y = x sinx logx
We can use the formula
\(\frac{d}{dx}\{{{f}_{1}}(x)\times {{f}_{2}}(x)\times {{f}_{3}}(x)\}={{f}_{2}}(x){{f}_{3}}(x)(\frac{d}{dx}{{f}_{1}}(x)+{{f}_{1}}(x){{f}_{3}}(x)\frac{d}{dx}{{f}_{2}}(x)+{{f}_{1}}(x){{f}_{2}}(x)\frac{d}{dx}{{f}_{3}}(x)\),
y’ = d/dx (x sinx logx)
= sinx logx d/dx(x) + x logx d/dx(sinx) + x sinx d/dx(logx)
= sinx logx + x logx (cosx) + x sinx (1/x)
y’ = sinx logx + x logx cosx + sinx