Squaring and differentiating again we get, Differentiating n times using Leibnitz theorem, (1 – x2)yn+2 – (2n + 1)xyn+1 – (n2 – m2) yn = 0.Weather - The Weather Channel 4+ - App Store. Now differentiating each term n times by Leibnitz theorem, we get + + n2yn = 0 x2 yn+2 + (2n + 1) xyn+1 + 2n2 yn = 0
Therefore the theorem is true for m + 1 and hence by the principle of mathematical induction, the theorem is true for any positive integer n.Įxample: If y = sin (m sin-1 x) then prove that (i) (1 – x2) y2 – xy1 + m2 y = 0 (ii) (1 – x2) yn+2 – (2n + 1) xyn+1 + (m2 – n2) yn = 0. Step 1: Take n = 1 By direct differentiation, (uv)1 = uv1 + u1vįor n = 2, (uv)2 = u2v+ u1v1 + u1v1+ uv2 Step 2: We assume that the theorem is true for n = m Differentiating both sides we get Leibnitz’s Theorem: If u and v are functions of x possessing derivatives of the nth order, then Proof: The Proof is by the principle of mathematical induction on n. Find the nth derivative of y = cos h2 3x Solution: Write cos h2 3x =įind the nth derivative of : (1) sin h 2x sin 4x Solution: Dn (2) y = log (4x2 – 1) Solution: Let y = log (4x2 – 1) = log Therefore y = log (2x + 1) + log (2x – 1). Y1 = eax Note: sin (A + B) = sin A cos B + cos A sin B y1 = r eax sin ( + bx + c) Similarly we get, y2 = r2 eax sin (2 + bx + c), 圓 = r3 eax sin (3 + bx + c) yn = rn eax sin (n + bx + c)Įxercise: If y = eax cos (bx + c), yn = rn eax cos (n + bx + c), Examples: 1. b cos (bx + c) + aeax sin (bx + c), = eax Put a = r cos , b = r sin Then = tan-1 (b/a) and a2 + b2 = r2 (cos2 + sin2 ) = r2 y = (ax + b)m, where m is a positive integer such that m > n. (m amx log a) = (m log a)2 amx 圓 = (m log a)3 amx yn = (m log a)n amx.ģ. Nth order derivatives of some standard functions: 1. These two processes act inversely to each other, a fact delivered conclusively by the Fundamental theorem of calculus. The second is the integral calculus ( Part B ), which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. More precisely, the local behavior of functions which can be illustrated by the slope of a function's graph.
The first is the differential calculus ( Part A ), which is concerned with the instantaneous rate of change of quantities with respect to other quantities.
It is built on two major complementary ideas, both of which rely critically on the concept of limits. ANONYMUS Calculus is a central branch of Mathematics, developed from algebra and geometry. Differential Calculus To err is human, to admit superhuman, to forgive divine, to blame it on others politics, to repeat unprofessional.
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