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,
Find the derivative of each of the following.
Problem 1 :
tan(7x)
Solution :
We already know the derivative of tanx, which is sec^{2}x. We can find the derivative of tan(7x) using chain rule.
= [tan(7x)]'
= [sec^{2}(7x)](7x)'
= [sec^{2}(7x)](7)
= 7sec^{2}(7x)
Problem 2 :
tan(4x - 13)
Solution :
= [tan(4x - 13)]'
= [sec^{2}(4x - 13)](4x - 13)'
= [sec^{2}(4x - 13)](4 - 0)
= [sec^{2}(4x - 13)](4)
= 4sec^{2}(4x - 13)
Problem 3 :
tan(3x^{2} - x + 5)
Solution :
= [tan(3x^{2} - x + 5)]'
= [sec^{2}(3x^{2} - x + 5)](3x^{2} - x + 5)'
= [sec^{2}(3x^{2} - x + 5)](6x - 1 + 0)
= [sec^{2}(3x^{2} - x + 5)](6x - 1)
= (6x - 1)sec^{2}(3x^{2} - x + 5)
Problem 4 :
tan^{2}x
Solution :
= (tan^{2}x)'
= (2tan^{2-1}x)(tanx)'
= (2tanx)(sec^{2}x)
= 2tanxsec^{2}x
Problem 5 :
Solution :
Problem 6 :
tan√x
Solution :
Problem 7 :
e^{tanx}
Solution :
= (e^{tanx})'
= e^{tanx}(tanx)'
= e^{tanx}(sec^{2}x)
= (sec^{2}x)e^{tanx}
Problem 8 :
ln(tanx)
Solution :
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