Introduction to Derivatives

DERIVATIVE OF F(X) FROM THE FIRST PRINCIPLE

$\frac{dy}{dx}=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$ = f' (x)

The limit which is a function of x is called the derivative of f (x) and is denoted by f' (x).

The symbol f' (c), then denotes the value of the function f' (x) for x = c.

Example -1: Find the derivative of x³.

Solution: Let f (x) = x³

So f' (x) = $\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$ = $\underset{h\to 0}{\lim }\frac{{(x+h)}^3-x^3}{h}$

= $\underset{h\to 0}{\lim }{h}^2+3xh+3x^2$

$\frac{dy}{dx}$ = f' (x) = 3x².

Example -2: Find the derivative of $x^{\frac{5}{2}}$.

Solution: $\frac{d{\left(x\right)}^{5/2}}{dx}=\frac{5}{2}{x}^{3/2}$

FUNDAMENTAL RULES FOR DIFFERENTIATION

(i) Differentiation of a constant function = 0

i.e. $\frac{d}{dx}C=0$

(ii) Derivative of the sum: Let u, v be two derivable functions of x so denoting their sum by y, we write, y = u + v

so, $\frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}$

(iii). Derivative of the difference: Let u, v be two derivable function of x, so denoting their difference by y, so we write y = u –v

so, $\frac{dy}{dx}=\frac{du}{dx}-\frac{dv}{dx}$

(iv). Generalisation: By a repeated application of the results obtained above, it can be proved that if u₁, u₂ ……, u_n be any finite number of derivable functions, then

y = u₁ $\pm$ u₂ $\pm$ u₃ $\pm$ ……. $\pm$ u_n

So, $\frac{dy}{dx}=\frac{d{u}_1}{dx}\pm \frac{d{u}_2}{dx}\pm \frac{d{u}_3}{dx}\pm .......\pm \frac{d{u}_n}{dx}$

$\Rightarrow$ $\frac{dy}{dx}=\left(u_2{u}_3....u_n\right)\frac{d{u}_1}{dx}+\left(u_1{u}_3....u_n\right)\frac{d{u}_2}{dx}+....+\left(u_1{u}_2....u_{n-1}\right)\frac{d{u}_n}{dx}$

Example -3: Find $\frac{dy}{dx}$, if y = $\frac{x^2+1}{x}$.

Solution: Let y = x + $\frac{1}{x}$, so let u = x₁ and v = $\frac{1}{x}$

So y = u + v. = 1 –$\frac{1}{x^2}$ = $\frac{x^2-1}{x^2}$

Example -4: Find $\frac{dy}{dx}$, if y = $\frac{x^3-1}{x}$.

Solution: Let y = x² – $\frac{1}{x}$, so let u = x² and v = $\frac{1}{x}$

So y = u – v

$\frac{dy}{dx}=\frac{du}{dx}-\frac{dv}{dx}$ = 2x +$\frac{1}{x^2}$ = $\frac{2x^2+1}{x^2}$