# Summary

**Basic laws of differentiation***f*(*x*) and*g*(*x*) be any two functions of*x*. Then,- Scalar multiple rule
*c*is any constant - Sum and difference rule
- Product rule
- Quotient rule
**Derivative of a function of a function***y*=*f*(*u*), where*u*=*g*(*x*)**Implicit functions***f*(*x*,*y*) = 0 is said to be an implicit function, if*y*cannot be directly defined as a function of*x*.*x*, collect the terms containing on one side; transfer other terms to the other side and divide by the coefficient of to get its value.**Parametric equation***x*=*f*(*t*) and*y*=*g*(*t*), then**Logarithmic differentiation****Higher order derivatives***y*=*f*(*x*), be a function of*x*. is called the first derivative of*y*with respect to*x*.*f*â€²(*x*) is called the second derivative of*y*with respect to*x*,*i.e.*,**Integral Calculus***y*with respect to*x*is given by , the integral of*f*â€²(*x*) with respect to*x*is given by**Basic laws of integration***f*(*x*) and*g*(*x*) be any two functions of*x*, then

# Integration by Substitution

Sometimes it is not possible to be able to integrate*f*(

*x*) directly. We can substitute

*f*(

*x*) into some other function

*g*(

*t*) to make it readily integrable.

** **

** **

# Integration by Parts

Let*u*=

*f*(

*x*) and

*v*=

*g*(

*x*) be two different functions of

*x*. Then

The function that can be easily integrated should be chosen as *v* and the other function which is easily differentiable should be chosen as *u*.

# Integration by the Method of Partial Fraction

**Type 1:**

** **

where *A* and *B* are constants to be determined.

**Type 2:**

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where *A*, *B* and *C* are constants to be determined.

**Type 3:**

** **

where *A* and *B* and *C* are constants to be determined.

# Definite Integration

Let where*f*(

*x*) is the integral of

*F*(

*x*).

As *x* changes from *a* to *b*, the value of the integral changes from *f *(*a*) to *f *(*b*). This can be shown as

# Properties of Definite Integrals

- is an even function
- if
*f*(*x*) is an odd function