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 […]
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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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A series of the from is called power series expansion of a function , where are constants and is variable. Mathematics Of Power Series Expansion: In order to explore power series we have to recall that the sum of a geometric series can be expressed using the simple formula: If , and that the series diverges when . At
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Vector Triple Product Definition: Consider be the three vectors, then the vector triple product of vectors is defined as Important Note: Through given any three vectors the following products are the vector triple products : Employing the well-known properties of the vector product, we get the following theorems. Theorem 1 : It satisfies the
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