DERIVATIVE OF TANX USING QUOTIENT RULE

The tangent of x is defined as sine of x divided by cosine of x.

tanx = ˢⁱⁿˣ⁄cₒsₓ

Since tanx can be written as sinx divided by cosx, we can find the derivative of tanx using quotient rule.

Derivative of Tanx :

Substitute tanx for y, sinx for u and cosx for v into the above formula and and find the derivative of tanx.

Therefore, the derivative of tangent of x is sec²x.

Solved Problems

Find the derivative of each of the following.

Problem 1 :

tan(3x)

Solution :

We already know the derivative of tanx, which is sec²x. We can find the derivative of tan(3x) using chain rule.

= [tan(3x)]'

= [sec²(3x)](3x)'

= [sec²(3x)](3)

= 3sec²(3x)

Problem 2 :

tan(5x - 6)

Solution :

= [tan(5x - 6)]'

= [sec²(5x - 6)](5x - 6)'

= [sec²(5x - 6)](5)

= 5sec²(5x - 6)

Problem 3 :

tan(2x² - 3x + 1)

Solution :

= [tan(2x² - 3x + 1)]'

= [sec²(2x² - 3x + 1)](2x² - 3x + 1)'

= [sec²(2x² - 3x + 1)](4x - 3)

= (4x - 3)sec²(2x² - 3x + 1)

Problem 4 :

tan²x

Solution :

= (tan²x)'

= (2tan^2-1x)(tanx)'

= (2tanx)(sec²x)

Problem 5 :

Solution :

Problem 6 :

tan√x

Solution :

Problem 7 :

e^tanx

Solution :

= (e^tanx)'

= e^tanx(tanx)'

= e^tanx(sec²x)

= (sec²x)e^tanx

Problem 8 :

ln(tanx)

Solution :

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