# DERIVATIVE OF TANX USING QUOTIENT RULE

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

tanx = ˢⁱⁿˣ⁄cs

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

## Quotient Rule of Derivative

Consider a function defined by y as shown below.

y = ᵘ⁄ᵥ

Let u and v be the functions of x.

Then the derivative of y with respect to x :

The above formula is called the Quotient Rule of Derivative.

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)'

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

= (2tanx)(sec2x)

Problem 5 :

Solution :

Problem 6 :

tan√x

Solution :

Problem 7 :

etanx

Solution :

= (etanx)'

= etanx(tanx)'

= etanx(sec2x)

= (sec2x)etanx

Problem 8 :

ln(tanx)

Solution :

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