Using mathematical induction prove that d - dx x n - n x n -1 for all positive integers n-

Let P(n)=d/dx(x^n)=nx^n 1 Now, P(1) is d/dx(x^1)=1.x^1 1 e.e.,d/dx(x)=1, which is true. ∴ P(1) is true. Let P(m) be true, mϵ N ⇒ d/dx(x^m)=mx^m 1 ⇒ d/dx( ...

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Standard XII

Mathematics

Statement

Using mathema...

Question

Using mathematical induction prove that $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ for all positive integers n.

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$L e t P (n) = \frac{d}{d x} (x^{n}) = n x^{n - 1} N o w, P (1) i s \frac{d}{d x} (x^{1}) = 1. x^{1 - 1} e . e ., \frac{d}{d x} (x) = 1, w h i c h i s t r u e . ∴ P (1) i s t r u e . L e t P (m) b e t r u e, m ϵ N \Rightarrow \frac{d}{d x} (x^{m}) = m x^{m - 1} \Rightarrow \frac{d}{d x} (x^{m + 1}) = \frac{d}{d x} {x^{m} x} = x^{m} \frac{d}{d x} (x) + x \frac{d}{d x} (x^{m}) (b y p r o d u c t r u l e) = x^{m} .1 + x (m x^{m - 1}) = (m + 1) x^{m} [u s i n g E q . (i)] \Rightarrow \frac{d}{d x} (x^{m + 1}) = {m + 1} x^{(m + 1) - 1} \Rightarrow P (m + 1) i s t r u e . T h u s, P (1) i s t r u e a n d P (m) \Rightarrow P (m + 1), m ϵ N . H e n c e, b y i n d u c t i o n P (n) i s t r u e f o r a l l n ϵ N$

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Using mathematical induction prove that ddx(xn)=nxn−1 for all positive integers n.

Using mathematical induction prove that $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ for all positive integers n.