Derivative of e to the Power x by First Principle

Let

f(x) = e^x

Derivative of cosx using first principle :

Solved Problems

Find the derivative of each of the following.

Problem 1 :

e^2x

Solution :

We already know the derivative of e^x, which is e^x. We can find the derivative of e^2x using chain rule.

= [e^2x]'

= [e^2x](2x)'

= [e^2x](2)

= 2e^2x

Problem 2 :

e^{3x + 5}

Solution :

= [e^{3x + 5}]'

= [e^{3x + 5}](3x + 5)'

= [e^{3x + 5}](3 + 0)

= [e^{3x + 5}](3)

= 3e^{3x + 5}

Problem 3 :

Solution :

Problem 4 :

e^lnx

Solution :

= (e^lnx)'

= e^lnx(lnx)'

= e^lnx(¹⁄ₓ)

= (¹⁄ₓ)e^lnx

= (¹⁄ₓ)(x)

= 1

Problem 5 :

e^sinx

Solution :

= (e^sinx)'

= e^sinx(sinx)'

= e^sinx(cosx)

= cosxe^sinx

Problem 6 :

e^cosx

Solution :

= (e^cosx)'

= e^cosx(cosx)'

= e^cosx(-sinx)

= -sinxe^cosx

Problem 7 :

e^tanx

Solution :

= (e^tanx)'

= e^tanx(tanx)'

= e^tanx(sec²x)

= sec²xe^tanx

Problem 8 :

e^cscx

Solution :

= (e^cscx)'

= e^cscx(cscx)'

= e^cscx(-cscxcotx)

= -cscxcotxe^cscx

Problem 9 :

e^secx

Solution :

= (e^secx)'

= e^secx(secx)'

= e^secx(secxtanx)

= secxtanxe^secx

Problem 10 :

e^cotx

Solution :

= (e^cotx)'

= e^cotx(cotx)'

= e^cotx(-csc²x)

= -csc²xe^cotx

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