15. The Derivative

Dated: 30-10-2024

We know that

\[m_{tan} = \lim_{x_1 \rightarrow x_0} \frac{f(x_1) - f(x_0)}{x_1 - x_0}\]

Let us define a constant \(h = x_1 - x_0\) such that \(x_1 = h + x_0\).
Substitute values and we will get

\[m_{tan} = \frac{dy}{dx} = \lim_{h \rightarrow 0} \frac{f(h + x_0) - f(x_0)}{h}\]

Equation of Tangent line

Following is the equation of tangent line at a point \(P(x_0, x_1)\).

\[y - y_0 = m(x - x_0)\]

We first find a \(m\) from the derivative equation and then plug the values in to get the equation for tangent.

The process of finding the derivative is called differentiation.
The derivative of \(f(x)\) with respect to \(x\) can be written as.

\[\frac{d}{dx}\left(f(x)\right) = f^{\prime}(x)\]

\[\left. \frac{d}{dx}\sqrt{x} \right|_{x = x_0} = \left. \frac{1}{2 \sqrt{x}} \right|_{x = x_0} = \frac{1}{2 \sqrt{x_0}} \]

There are some situations when we can say that the derivative does not exist at a certain point for a function.¹