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Chapter 2: Q. 5 (page 221)

Explain how the formula for differentiating the natural exponential function is a special case of the formula for differentiating exponential functions of the form ekx. Then explain why it is a special case of the formula for differentiating functions of the form bx.

Short Answer

Expert verified

For ekxwe take k and e as constants so it is a form ofeax

For bxwe take b=e because it is a special case of the formex

Step by step solution

01

. Given Information

We are given two forms :ekxandbx

02

Step 2. Differentiating ekx 

We differentiate the form :

ddxekx=kekx,kandeareconstants

03

. Differentiating bx

We differentiate the form :

ddxbx=bxlogeb

We use b=e because it is a special case, so we get :

ddxex=exlogee=ex

04

. Explaining

So we can say the ekxis a form of eaxand bxis a special form ofex

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