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Chapter 2: Q. 83 (page 235)

Use the definition of the derivative, a trigonometric identity, and known trigonometric limits to prove that ddx(cosx)=-sinx

Short Answer

Expert verified

The trigonometric limit ddx(cosx)=-sinx has been proved.

Step by step solution

01

Step 1. Given Information.

The trigonometric identity:

ddx(cosx)=-sinx

02

Step 2. Definition of derivative.

From the definition derivative,

f'(x)=limh鈫�0f(x+h)-f(x)h

03

Step 3. Obtain the derivative for the given function.

From the definition of derivative,

f'(cosx)=limh鈫�0[cos(x+h)-cosxh]=limh鈫�0[(cosxcosh-sinx.sinh)-cosxh]=limh鈫�0[cosx(cosh-1)-sinxsinhh]=limh鈫�0[cosxcosh-1h-sinxsinhh]

04

Step 4. Apply the limits.

So the function become,

f'(cosx)=cosxlimh鈫�0cosh-1h-sinxlimh鈫�0sinhh=cosx(0)-sinx(1)=-sinx

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