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DIFFERENTIATION AND INTEGRATION
Section 1: Differentiation
Definition of derivative
A derivative $f^{\prime}(x)$ of a function $f(\mathrm{x})$ depicts how the function $f(\mathrm{x})$ is changing at the point ' x '. It is necessary for the function to be continuous at the point ' x ' for the derivative to exist. A function that has a derivative is said to be differentiable. In general, the derivative of the function $\mathrm{y}=f(\mathrm{x})$, also denoted $\mathrm{dy} / \mathrm{dx}$, can be defined as
\[
\frac{d y}{d x}=\frac{d f(x)}{d x}=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}
\]
This means that as $\Delta \mathrm{x}$ gets very small, the difference between the value of the function at ' x ' and the value of the function at $\mathrm{x}+\Delta \mathrm{x}$ divided by $\Delta \mathrm{x}$ is defined as the derivative.
Derivatives of some common functions
\[
\begin{array}{l}
\frac{d}{d x}(x)=1
\frac{d}{d t}\left(t^{2}\right)=2 t
\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}
\frac{d}{d x}(\log x)=\frac{1}{x}
\frac{d}{d x}\left(e^{a x}\right)=a e^{a x}
\frac{d}{d x}(\sin x)=\cos x
\frac{d}{d x}(\cos x)=-\sin x
\end{array}
\]
General rules of differentiation
1. The derivative of a constant is equal to zero. If $y=c$,
\[
\frac{d y}{d x}=\frac{d}{d x}(c)=0
\]
where ' c ' is any arbitrary constant.
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