Derivative of Tangent
Derivative of the tangent function.
Formula
Variables
Example
The slope of tan x at x = 0 is sec²0 = 1.
How it works
The derivative of tan x tells you the instantaneous rate at which the tangent function changes as x changes. It comes from writing tan x as sin x / cos x and applying the quotient rule; after simplifying, the result is sec²x. Since sec x = 1/cos x, this also shows the slope grows very quickly near angles where cos x is close to 0.
When to use it
Use this rule whenever you need the slope of a curve involving tan x, or when differentiating expressions that contain tangent as part of a larger function. It is especially useful in physics, engineering, and calculus problems involving angles, periodic behavior, and trig-based models.
Worked example, step by step
Find the derivative of f(x) = 5tan x - 3 f'(x) = 5(d/dx)(tan x) - d/dx(3) f'(x) = 5sec²x - 0 f'(x) = 5sec²x So the derivative is 5sec²x.
Common mistakes to avoid
- Writing the derivative of tan x as sec x instead of sec²x
- Forgetting the chain rule when the input is something like tan(3x)
- Confusing sec²x with (sec x)² and dropping the square entirely
Did You Know?
The derivative of tan x is sec²x, which is never negative — tangent is always increasing between its asymptotes.
Share this formula
More in Calculus
View allPower Rule (Derivative)
IntermediateDerivative of a power function.
Chain Rule
IntermediateDerivative of a composite function.
Product Rule
IntermediateDerivative of a product of two functions.
Quotient Rule
IntermediateDerivative of a quotient of two functions.
Derivative of Sine
IntermediateDerivative of the sine function.
Derivative of Cosine
IntermediateDerivative of the cosine function.