DERIVATIVE OF TANX BY FIRST PRINCIPLE

The formula to find derivative of a function f(x) using first principle :

Let

f(x) = tanx

Derivative of tanx using first principle :

From one of the Trigonometric Identities,

sinAcosB - cosAsinB = sin(A - B)


From standard results of limits,


Solved Problems

Find the derivative of each of the following.

Problem 1 :

tan(7x)

Solution :

We already know the derivative of tanx, which is sec2x. We can find the derivative of tan(7x) using chain rule.

= [tan(7x)]'

= [sec2(7x)](7x)'

= [sec2(7x)](7)

= 7sec2(7x)

Problem 2 :

tan(4x - 13)

Solution :

= [tan(4x - 13)]'

= [sec2(4x - 13)](4x - 13)'

= [sec2(4x - 13)](4 - 0)

= [sec2(4x - 13)](4)

= 4sec2(4x - 13)

Problem 3 :

tan(3x2 - x + 5)

Solution :

= [tan(3x2 - x + 5)]'

= [sec2(3x2 - x + 5)](3x2 - x + 5)'

= [sec2(3x2 - x + 5)](6x - 1 + 0)

= [sec2(3x2 - x + 5)](6x - 1)

= (6x - 1)sec2(3x2 - x + 5)

Problem 4 :

tan2x

Solution :

= (tan2x)'

= (2tan2-1x)(tanx)'

= (2tanx)(sec2x)

= 2tanxsec2x

Problem 5 :

Solution :

Problem 6 :

tanx

Solution :

Problem 7 :

etanx

Solution :

= (etanx)'

= etanx(tanx)'

= etanx(sec2x)

= (sec2x)etanx

Problem 8 :

ln(tanx)

Solution :

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