DERIVATIVE OF COTX USING QUOTIENT RULE

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

cotx = ᶜᵒˢˣ⁄sᵢnₓ

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

Derivative of Cotx :

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

Therefore, the derivative of cotangent of x is -csc²x.

Solved Problems

Find the derivative of each of the following.

Problem 1 :

cot(2x)

Solution :

We already know the derivative of cotx, which is -csc²x. We can find the derivative of cot(2x) using chain rule.

= [cot(2x)]'

= [-csc²(2x)](2x)'

= [-csc²(2x)](2)

= -2csc²(2x)

Problem 2 :

cot(3x + 5)

Solution :

= [cot(3x + 5)]'

= [-csc²(3x + 5)](3x + 5)'

= [-csc²(3x + 5)](3)

= -3csc²(3x + 5)

Problem 3 :

cot(x² + 3x + 5)

Solution :

= [cot(x² + 3x + 5)]'

= [-csc²(x² + 3x + 5)](x² + 3x + 5)'

= [-csc²(x² + 3x + 5)](2x + 3)

= -(2x + 3)csc²(x² + 3x + 5)

Problem 4 :

cot²x

Solution :

= (cot²x)'

= (2cot^2-1x)(cotx)'

= (2cotx)(-csc²x)

= -2(cotx)(csc²x)

Problem 5 :

Solution :

Problem 6 :

cot√x

Solution :

Problem 7 :

e^cotx

Solution :

= (e^cotx)'

= e^cotx(cotx)'

= e^cotx(-csc²x)

= -(csc²x)e^cotx

Problem 8 :

ln(cotx)

Solution :

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