# Theorems on Derivatives

## 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