# Fundamental rules for differentiation

Rule (I): Differentiation of a constant function is zero i.e., d/dx (c) = 0

Rule (II): Let f(x) be a differentiable function and let c be a constant. Then c.f(x) is also differentiable such that $$\frac{d}{dx}\left\{ c.f\left( x \right) \right\}=c.\frac{d}{dx}\left( f\left( x \right) \right)$$,

This is the derivative of a constant times a function is the constant times the derivative of the function.

Rule (III): If f(x) and g(x) are differentiable functions, then show that f(x) ± g(x) are also differentiable such that $$\frac{d}{dx}\left[ f\left( x \right)\pm g\left( x \right) \right]=\frac{d}{dx}f\left( x \right)\pm \frac{d}{dx}g\left( x \right)$$,

That is the derivative of the sum or difference of two functions is the sum or difference of their derivatives.

Rule (IV): If f(x) and g(x) are two differentiable functions, then f(x).g(x) is also differentiable such that $$\frac{d}{dx}\left[ f\left( x \right).g\left( x \right) \right]=f\left( x \right)\frac{d}{dx}\left[ g\left( x \right) \right]+\frac{d}{dx}\left[ f\left( x \right) \right].g\left( x \right)$$.

Rule (V) (Quotient rule): If f(x) and g(x) are two differentiable functions and g(x) ≠ 0 then f(x)/g(x) is also differentiable such that $$\frac{d}{dx}\left\{ \frac{f\left( x \right)}{g\left( x \right)} \right\}=\frac{g\left( x \right)\frac{d}{dx}\left[ f\left( x \right) \right]-f\left( x \right).\frac{d}{dx}\left[ g\left( x \right) \right]}{{{\left[ g\left( x \right) \right]}^{2}}}$$.

Relation between dy/dx and dx/dy: Let x and y be two variables connected by a relation of the form f(x, y) = 0. Let Δx be a small change in x and let Δy be the corresponding change in y Then $$\frac{dy}{dx}=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{\Delta y}{\Delta x}$$ and $$\frac{dx}{dy}=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{\Delta x}{\Delta y}$$.Now,

$$\frac{\Delta y}{\Delta x}.\frac{\Delta x}{\Delta y}=1$$,

⇒ $$\underset{\Delta x\to 0}{\mathop{\lim }}\,\left[ \frac{\Delta y}{\Delta x}.\frac{\Delta x}{\Delta y} \right]=1$$,

⇒ $$\frac{dy}{dx}.\frac{dx}{dy}=1$$ [∵ Δx → 0 ⇔ Δy → 0],

⇒ $$\frac{dy}{dx}.\frac{dx}{dy}=1$$,

Hence, $$\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}$$.

Ex: Differentiate f(x) = sinx.logx

Solution: Using product rule of differentiation

$$f’\left( x \right)=\frac{d}{dx}\left( \sin x \right).\log x+\sin x.\frac{d}{dx}\left( \log x \right)$$,

= logx.cosx + sinx/x

Ex: Find the derivative of f(x) = logx/x²

Solution: Using quotient rule of differentiation

$$f’\left( x \right)=\frac{{{x}^{2}}\frac{d}{dx}\left( \log x \right)-\log x\frac{d}{dx}\left( {{x}^{2}} \right)}{{{\left( {{x}^{2}} \right)}^{2}}}$$,

= $$\,\frac{{{x}^{2}}.\frac{1}{{{x}^{2}}}-\log x\left( 2x \right)}{{{x}^{4}}}$$,

= $$\,\frac{x-2x\log x}{{{x}^{4}}}$$,

= $$\,\frac{1-2\log x}{{{x}^{3}}}$$.