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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Statement of Mean Value Theorem (MVT) : Let a function be then there exist a point such that MVT Proof: Define a new function by where A is a constant to determined such that The functiom is contineous on and differentiable on . Hence is contineous and differentiable respecctively on and . The function satisfies
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What are the corresponding angles? When a third line intersects two parallel lines, the angle made by the same relative position at each intersection is called the corresponding angle to each other. When a line known as intersecting traversal crosses a pair of straight lines, corresponding angles are formed. Corresponding angles are the pair of
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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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