Mastering the Quotient Rule: Derivative of Tangent Function and Key Identities

Derivative of tan x

To find the derivative of the tangent function, we can use the quotient rule

To find the derivative of the tangent function, we can use the quotient rule.

The tangent function can be expressed as:
tan(x) = sin(x) / cos(x)

Using the quotient rule, the derivative of tan(x) with respect to x is given by:
d/dx[tan(x)] = (cos(x)*(d/dx[sin(x)]) – sin(x)*(d/dx[cos(x)])) / cos(x)^2

To find the derivatives of sin(x) and cos(x), we can use the following identities:

– The derivative of sin(x) is cos(x): d/dx[sin(x)] = cos(x)
– The derivative of cos(x) is -sin(x): d/dx[cos(x)] = -sin(x)

Substituting these derivatives into the quotient rule, we get:

d/dx[tan(x)] = (cos(x)*(cos(x)) – sin(x)*(-sin(x))) / cos(x)^2

= (cos^2(x) + sin^2(x)) / cos^2(x)

Now, using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify the expression:

d/dx[tan(x)] = 1 / cos^2(x)

Since tan(x) is not defined when cos(x) = 0 (at x = π/2, 3π/2, etc.), the derivative of tan(x) is not defined at those points as well. In other words, the derivative of tan(x) is undefined when cos(x) = 0.

In conclusion, the derivative of tan(x) is 1 / cos^2(x) except at points where cos(x) = 0.

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