Quotient Rule Derivative

Quotient Rule Derivative Quotient Rule Derivative n mathematics, quotient rule is basically a part of calculus. It is an efficient method to find the derivative of any function. It is a simple or normal rule to find the differentiation of the given problem in which one function gets divided by another function. It means that the quotient of any two other functions for which it derivatives must exist .let a and b are the differentiable functions.

Suppose that we have to differentiate a function f(y), then it can be written as: f(y) = a(y) / b(y) where b(y) ≠ 0 ,so according to the quotient rule the derivative of a(y) / b(y) is given by: f’(y) = a’(y) / b(y) - a(y) / b’(y) / [ b(y)]2 therefore, suppose that y satisfies all the open set which contains the number i.e. b(y) ≠ 0 and a’(b) and b’(u) all exists, then f’(v) also exists, so f’(v) = a(v) / b’(v) – a’(v) / b(v) / [ b(v)]2

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the above expression can also be use to calculate the 2nd derivative, then it can be written as: f’’(y) = a” (y) [b(y)]2 – 2a’ (y) b(y) b’(y) + a(x) [2[ b’(y) ]2 – b(y) b’’(y)] / [b(y)]3 Product rule and chain rule is the derivative of the Quotient Rule.

Product rule of a function is given by: f(y) . a(y) = [f’(y) . a(y)] + (f(y) . a’(x)] we can also derive the expression of quotient rule through chain rule: suppose, p / q = ¼[(p + 1 / q)2 – (p – 1 / q)2] now, d(p / q)/dy = d/dy . ¼[(p+1 / q)2 – (p – 1 / q)2] d(p / q)/dy = ¼ [2(p + 1/q) (dp /dy – dq /q2. dy) – 2(p – 1 / q) (dp /dy + dq / q2. dy)] multiplying the above terms we get: d(p / q) / dy = ¼[4 / q . dp / dy – 4p / q2 . dq / dy] at last the expression is written as: d(p /q)/dy = [q.dp /dy – p.dq /dy] /q2 there are certain limitations of the quotient rule in which the numerator or denominator are not differentiable but it may be possible that the quotient may be differentiable.

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Suppose a function is given by : f(y) = |y| + 1 / |y|+ 1 in the given above equation |y| is the absolute value of y. This is the limitation of the quotient rule. lets deal the quotient rule with an example: If a given function f (y) = 3y + 1/ y - 2, then calculate the quotient rule?

Solution: f’(y) = y – 2 . d/dy . [3y + 1] – (3y + 1).d/dy.[y - 2] / [y - 2]2 = y – 2 . (3) – (3y + 1)(1) / [y - 2]2 = - 7 / [y - 2]2 Now we will have a little description about the bilateral Laplace transform.

Bilateral Laplace transform is also known as two sided Laplace transform. The expression of a bilateral Laplace transform is given by: F(u) = B f(v) = ò-∞∞ e-uv. f(v) . dv.

In bilateral Laplace transform, we have to just extend the limit i.e. from -∞ to ∞ whereas in Laplace transform limit extends from 0 to ∞. The only difference between them is of limits.

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