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Q. 86

Found in: Page 199


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Prove, in two ways, that the power rule holds for negative integer powers

a) by using the zx definition of the derivative

b) by using the h0 definition of the derivative

We prove the power rule for negative powers.

See the step by step solution

Step by Step Solution

Step 1: Given information

We are given a function f(x)=x-n

Step 2: Find the derivative using z→x

We get,

limzxf(z)-f(x)z-xlimzxz-n-x-nz-xlimzxzk-xkz-x [Replace -n by k]limzx(z-k)(zk-1+zk-2x+....+xk-1)z-xlimzx(zk-1+zk-2x+....+xk-1)=xk-1+xk-1+.....+xk-1=kxk-1Now replace k by -n=-nx-n-1

Step 3: Using h→0

We get,

limh0f(x+h)-f(x)hlimh0(x+h)-n-x-nhReplace -n by klimh0(x+h)k-xkhlimh0xk+kxk-1h+....+hk-xkhlimh0kxk-1h+....+hkh=kxk-1Now replace k by -n=-nx-n-1

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