# Derivative of e to the Power x by First Principle

Formula to find derivative of a function f(x) by first principle :

Let

f(x) = ex

Derivative of cosx using first principle :

From standard results of limits,

## Solved Problems

Find the derivative of each of the following.

Problem 1 :

e2x

Solution :

We already know the derivative of ex, which is ex. We can find the derivative of e2x using chain rule.

= [e2x]'

= [e2x](2x)'

= [e2x](2)

= 2e2x

Problem 2 :

e3x + 5

Solution :

= [e3x + 5]'

= [e3x + 5](3x + 5)'

= [e3x + 5](3 + 0)

= [e3x + 5](3)

= 3e3x + 5

Problem 3 :

Solution :

Problem 4 :

elnx

Solution :

= (elnx)'

= elnx(lnx)'

= elnx(¹⁄ₓ)

= (¹⁄ₓ)elnx

By Identity,

elnx = x

= (¹⁄ₓ)(x)

= 1

Problem 5 :

esinx

Solution :

= (esinx)'

= esinx(sinx)'

= esinx(cosx)

= cosxesinx

Problem 6 :

ecosx

Solution :

= (ecosx)'

= ecosx(cosx)'

= ecosx(-sinx)

= -sinxecosx

Problem 7 :

etanx

Solution :

= (etanx)'

= etanx(tanx)'

= etanx(sec2x)

= sec2xetanx

Problem 8 :

ecscx

Solution :

= (ecscx)'

= ecscx(cscx)'

= ecscx(-cscxcotx)

= -cscxcotxecscx

Problem 9 :

esecx

Solution :

= (esecx)'

= esecx(secx)'

= esecx(secxtanx)

= secxtanxesecx

Problem 10 :

ecotx

Solution :

= (ecotx)'

= ecotx(cotx)'

= ecotx(-csc2x)

= -csc2xecotx

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