Here, we can prove and learn differentiation formulas like the product rule, power rule, Quotient rule, trigonometric function derivative, constant derivative and much more. All proofs are available below. 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 […]
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Binomial Theorem: where and are real numbers and are binomial cofficient. and are exponents and is called index. The exponent of decreases from index to zero. The exponent of increases from zero to index. The degree of each terms is equal to its index. The numbers of terms in the expansion is one greater than
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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
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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
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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
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Rolle’s Theorem Statement: let a function be 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 If then the funtion is constant on so its derivatives vanishes
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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
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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
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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.
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Completing the Square examples: Sometimes, the quadratic polynomials are not easily factorable. For Example, consider It is difficult to make factors of In such a case the factorization and hence the solution of a quadratic equation can be found by the method of completing the square and extracting square roots. Completing the Square Examples Say
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