Differentiation
1.0 Differentiation
1.0 Differentiation
The differentiation/derivative is the instantaneous rate of change of a function with respect to one of its variables. It is denoted by, $$\frac{dy}{dx}\ or\ f ' (x)$$
The derivative of any function $f(x)$ with respect to $x$ is defined as,
$$f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}$$
Example 1. Differentiate $y = {x^2} + 7x$
Solution: We know that, $$f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}$$
$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{h \to 0} \frac{{\left( {{{\left( {x + h} \right)}^2} + 7\left( {x + h} \right)} \right) - \left( {{x^2} + 7x} \right)}}{h}$$
Using Identity,
$$\begin{equation} \begin{aligned} {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab \\ \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{h \to 0} \frac{{\left( {{x^2} + {h^2} + 2xh + 7x + 7h} \right) - \left( {{x^2} + 7x} \right)}}{h} \\ \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{h \to 0} \frac{{\left( {{h^2} + 2xh + 7h} \right)}}{h} \\ \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{h \to 0} \frac{{h\left( {h + 2x + 7} \right)}}{h} \\ \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{h \to 0} \left( {h + 2x + 7} \right) = 0 + 2x + 7 = 2x + 7 \\\end{aligned} \end{equation} $$
