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Calculus Formulas

Calculus is a branch of mathematics which is based on the differential calculus and integral calculus. Differential calculus (differentiation), is known as the mathematics of change, and is used for determining the slope of a line tangent to a curve at a point on the curve. Integral Calculus (integration) is the process of measuring small changes in value, its opposite, the process of summing the intervals under a curve.

Mathematicians have developed certain rules for finding the derivative and integration of a given function. 
The most commonly used calculus formulas where, c is a constant, and u and v are the functions of x are listed below:

For Differentiation:
  1. Derivative of any constant is zero. That is, $\frac{d}{dx}$$(c)$ = 0 
  2. Derivative of x with respect to x is 1. That is, $\frac{d}{dx}$$(x)$ = 1
  3. $\frac{d}{dx}$$(u + v)$ = $\frac{du}{dx}$ + $\frac{dv}{dx}$ 
  4. $\frac{d}{dx}$$(u - v)$ = $\frac{du}{dx}$ - $\frac{dv}{dx}$ 
  5. Product Rule: $\frac{d}{dx}$$(u \times v)$ = u x $\frac{dv}{dx}$ + v x $\frac{du}{dx}$ 
  6. Quotient Rule: $\frac{d}{dx}\frac{u}{v}$ = $\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$
  7. Power Rule: $\frac{d}{dx}$$u^{n}$ = $n u ^{n-1}$ $\frac{du}{dx}$
  8. A derivative of the logarithm is $\frac{d}{dx}$$(\log u)$ = $\frac{1}{u} \times \frac{du}{dx}$
  9. A derivative of the exponential function is $\frac{d}{dx}$$e^{u}$ = $e^{u}$$\frac{du}{dx}$.
For Integration:

  1. $\int (dx)= x + c$
  2. $\int (u+v) dx= \int udx + \int vdx +c$
  3. $\int (u-v) dx= \int udx - \int vdx +c$
  4. $\int u^{n}dx$ = $\frac{u^{n+1}}{n+1}$$+c$. Here, n is not equal to -1
  5. $\int $$\frac{1}{u}$$du = \log u + c$
  6. $\int e^{u}du = e^{u} + c$
Examples with Calculus Formulas:

Given below are some of the examples that makes use of calculus formulas.

Example 1: 

Find the derivative of the square of root of x.

Solution:

We know the derivative of $x^{n}$ is equal to n$x^{n-1}$.

Derivative of $\sqrt{x}$ is equal to the derivative of $x^{\frac{1}{2}}$.
This is equal to $\frac{1}{2}x^{\frac{-1}{2}}$.

Example 2:

Find the derivative of y = x sin x with respect to x.

Solution:

We can use the product formula: Derivative of (u v) = u v' + v u'.

$\frac{dy}{dx}$ = $\frac{d}{dx}$ (x sin x)

= x $\frac{d}{dx}$ sin x + sin x $\frac{d}{dx}$

= x cos x + sin x.

Example 3:

Find the integration of $\frac{1}{x \log x}$

Solution:

The integration of $\frac{1}{x \log x}$ is equal to the integration of $\frac{\frac{1}{x}}{\log x}$.

=>  $\int$ $\frac{1}{x \log x}$ dx = $\int$ $\frac{\frac{1}{x}}{\log x}$ dx

Substitute log x = t

Then, its derivative is $\frac{1}{x}$ dx = dt

Therefore, $\int$ $\frac{\frac{1}{x}}{\log x}$ dx = $\int$ $\frac{1}{t}$ dt

= log (t) + c, where c is the constant of integration.

Resubstituting the value of t, we get

$\int$ $\frac{1}{x \log x}$ = log |log x|+ c

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