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,
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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