Step-by-Step Guide to Computing the Derivative of the Tangent Function Using Quotient Rule and Chain Rule

d/dx(tanx)

sec²x

To compute the derivative of the tangent function, we start with the definition of tangent:

tan(x) = sin(x)/cos(x)

Then, we can use the quotient rule to find the derivative:

d/dx(tan(x)) = (cos(x) d/dx(sin(x)) – sin(x) d/dx(cos(x))) / (cos(x))^2

Next, we need to find the derivatives of sin(x) and cos(x):

d/dx(sin(x)) = cos(x)

d/dx(cos(x)) = -sin(x)

Substituting these derivatives back into the quotient rule formula, we get:

d/dx(tan(x)) = (cos(x)cos(x) + sin(x)sin(x)) / (cos(x))^2

Simplifying the numerator by applying trigonometric identity, we get:

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

Alternatively, we could have used the chain rule and the derivative of the inverse cosine to get to the same final result:

tan(x) = sin(x)/cos(x)

(cos(x))^2 tan(x) = sin(x)cos(x)

(cos(x))^2 d/dx(tan(x)) + 2cos(x)tan(x) = cos(x)sin(x) – sin(x)cos(x)

(cos(x))^2 d/dx(tan(x)) = sin(x)cos(x) – cos(x)sin(x)

(cos(x))^2 d/dx(tan(x)) = 0

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

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