Mathematics

Binomial expansion: are called meaningless when is negative or fraction and  are exponents and  is called index. index is always less then one. exponent is always less then one. This series is called Binomial series. Example: using Binomial expansion solve this: HERE using Binomial expansion formula This is required binomial series.

Maclaurin series expension. The expension of is called the Maclaurin series expension. The above expansion is called Maclaurin Theorem. Example Apply Maclaurin series expension. Add title The expension of is called the Maclaurin series expension. The above expansion is called Maclaurin Theorem. Example Apply Maclaurin series expension. Put above equations Substituting these values in the

A very important special type of relation is a function defined as below: Function in Math: If A and B are two sets the relation f:A\rightarrow B is called function if for every element of A there exist a unique element of B, and Dom f = A. f:A\rightarrow B which is read: f is

Rolle’s Theorem Statement: let a function be Continuous on the interval . Differentiable on the open interval . then there exist at least one point such that . Proof: since is contineous on , it is bounded their attains its bounds. Let and be the supremum and infimum of on . then their either or

Derivative of product rule or differentiation of product rule let where and are function of Derivative of quotient rule or differentiation of quotient rule where and are function of Derivative of power rule or differentiation of power rule Derivative of chain rule or differentiation of chain rule Let and Derivative of trigonometric functions or Differentiation

Prove that: Subtracting eq(1) from eq(2) . The power rule for differentiation was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the mid-17th century for rational power functions, which both were then used to derive the power rule for integrals as the inverse operation. Here we have discussed the proof of power rule

Logarithmic Differentiation: Let where both and are variables or function of , the derivative of can be obtained by taking natural logarithms of both of sides and the differentiating . Example now by usning property of ln differentiate w.r.t x property of ln and product rule Example derivative of lnx and graph differentiate w.r.t x

Explicit Differentiation Explicit function: If is easily expressed in term of the independent variables ,Then is called an Explicit function of . Symbolically it is written as  Examples: Procedure: Step (1) when  and are not amalgamated or Explicit we assumethat  is differentiable function of . Step (2) Differentiate both sides of eq w.r.t . Step (3) Solve the resulting

HOMOGENEOUS EQUATION: Each equation of the system of following linear equations . . . is always satisfied by , so such a system is always consistent. The solution of the above homogeneous equations is called the trivial solution. Any other solution of equations other than the trivial solution is called a non-trivial solution. The above