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What limit facts and trigonometric identities are used in

the proof thatddxsinx=cosx?

Short Answer

Expert verified

limit facts limh0(cosh-1)h=0&limh0sinhh=1and sum identity for the sine sin(a+b)=sinacosb+cosasinbare used in the proof ofddxsinx=cosx.

Step by step solution

01

Step 1. Given information. 

the given formula isddxsinx=cosx.

02

Step 2. proof.

Use the definition of the derivative to derivate f(x)=sinx.

ddxsinx=limh0sin(x+h)-sinxh

Apply sum identity for the sine ,sin(a+b)=sinacosb+cosasinb.

role="math" localid="1648766873735" ddxsinx=limh0sinxcosh+cosxsinh-sinxhddxsinx=limh0sinx(cosh-1)+cosxsinhhddxsinx=sinxlimh0(cosh-1)h+cosxlimh0sinhh

Use limit fact limh0(cosh-1)h=0&limh0sinhh=1

ddxsinx=sinx0+cosx1ddxsinx=cosx

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