4.0 Differentiation

Differentiation is the process of finding the derivative of a differentiable function.

The derivative of a function measures the rate at which the function value changes as its input changes.

The derivative of a function $y=f(x)$ is defined as,
$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}$$or$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {x + \Delta x} \right) - f(x)}}{{\Delta x}}$$

Question: Find the derivative of $y=x$ with respect to $x$.

Solution: $$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {x + \Delta x} \right) - f(x)}}{{\Delta x}}$$$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\left( {x + \Delta x} \right) - x}}{{\Delta x}}$$$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta x}}{{\Delta x}}$$$$\frac{{dy}}{{dx}} = 1$$

Question: Find the derivative of $y=x^2$ with respect to $x$.

Solution: $$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {x + \Delta x} \right) - f(x)}}{{\Delta x}}$$$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{{{\left( {x + \Delta x} \right)}^2} - {x^2}}}{{\Delta x}}$$$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\left[ {{x^2} + {{\left( {\Delta x} \right)}^2} + 2x\Delta x} \right] - {x^2}}}{{\Delta x}}$$$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{{{\left( {\Delta x} \right)}^2} + 2x\Delta x}}{{\Delta x}}$$$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta x\left[ {\Delta x + 2x} \right]}}{{\Delta x}}$$$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \left[ {\Delta x + 2x} \right]$$$$\frac{{dy}}{{dx}} = 2x$$