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Use (a) the h→0 definition of the derivative and then (b) the z→c definition of the derivative to find f'(c) for each function f and value x = c.

f(x)=tanx,x=0

Short Answer

Expert verified

(a) f'(c)=1

(b)f'(c)=1

Step by step solution

01

Part (a) Step 1. Given information.

Given function isf(x)=tanx

We have to findf'(c)

02

Part (a) Step 2. Find the f'(c)

We have to find the derivative of the function using h→0 definition,

Therefore,

limh0f(0+h)f(0)h=limh0tan(0+h)tan0h=limh0tan(h)0h=limh0sin(h)hcos(h)=limh0sin(h)hlimh01cos(h)=1

03

Part (b) Step 1. Find f'(c)

Find the derivate of the function using x→0 definition
Therefore,

limx0f(x)f(0)x0=limx0tanxtan0x=limx0tanxh=limx0sinxxcosx=limx0sinxxlimx01cosx=1

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