Calculus is the study of change, it basically analyses things that change and are a significant part of Mathematics. Calculus is a branch of mathematics focused on limits, integrals, derivatives, functions and infinite series. Integral calculus and Differential calculus are the two main branches of this topic. Differential calculus is concerned with the study of rates at which a quantity changes whereas, Integral calculus gives information about the accumulation of quantities. These branches are connected with each other in respect of fundamental theorem. It is stated that calculus was founded in the 17th century and since then its concepts have been applied in many sectors including engineering, science, economics, computer science, medicine and others. Calculus is referred as the part of modern Mathematics.

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f(x) = sin x ... (1)

f(x + h) = sin (x + h) .....(2)

Subtract equation (1) from equation (2), we get

f(x + h) - f(x) = sin(x + h) - sin x

= 2 cos ($\frac{2x+h}{2}$) sin ($\frac{h}{2}$) [ using identity sin A - sin B = 2 cos($\frac{A+B}{2}$) sin ($\frac{A-B}{2}$)

= 2 cos (x + $\frac{h}{2}$) sin ($\frac{h}{2}$)

$\frac{f(x + h) - f(x)}{h}$ = $\frac{1}{h}$ (2 cos (x + $\frac{h}{2}$) sin ($\frac{h}{2}$))

= $\frac{2 cos (x + \frac{h}{2}) sin (\frac{h}{2})}{2 \times \frac{h}{2}}$ [ Because $\frac{sin \frac{h}{2}}{\frac{h}{2}}$ ]

= cos(x + $\frac{h}{2}$)

$\lim_{h\rightarrow0}$ $\frac{f(x + h) - f(x)}{h}$ = $\lim_{h\rightarrow0}$ cos(x + $\frac{h}{2}$)

= cos x

=> Derivative of sin x is cos x.

Given: $\int$ $\frac{log\ x}{x}$ dx

Substitute log x = t, then $\frac{1}{x}$ dx = dt

Now $\int$ $\frac{log\ x}{x}$ dx = $\int$ t dt = $\frac{t^2}{2}$

After re-substituting the values, we have

$\frac{(log\ x)^2}{2}$

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