What Is the Derivative of Tangent? The Short Answer
If you’re looking for the single most important fact: what is the derivative of tangent — it is sec²(x). In other words, for any real number x (where x is not an odd multiple of π/2, because tan and sec are undefined there),
This result is one of the six standard trigonometric derivatives and a building block for hundreds of calculus problems. Knowing what is the derivative of tangent lets you handle everything from simple slope calculations to complex physics involving oscillatory motion. The domain is x ≠ π/2 + kπ for any integer k; at those points the function and its derivative have vertical asymptotes.
🔑 Key Takeaways
- The derivative of tan(x) is always sec²(x), valid for x ≠ π/2 + πk.
- Three equivalent forms: sec²(x), 1/cos²(x), 1+tan²(x).
- Derived from the quotient rule using sin(x)/cos(x).
- When the argument is a function of x, apply the chain rule: d/dx tan(g(x)) = sec²(g(x)) · g'(x).
Proving the Derivative of Tangent: Step by Step
To truly internalize what is the derivative of tangent, let’s derive it from scratch. This proof reinforces your understanding and prepares you for more advanced calculus. We’ll use both the quotient rule and an alternative method for a deeper perspective.
Proof Using the Quotient Rule
Remember that
We can differentiate using the quotient rule:
Here, let \( u = \sin x \) and \( v = \cos x \). Then \( u’ = \cos x \) and \( v’ = -\sin x \).
The numerator is the Pythagorean identity: \( \cos^2 x + \sin^2 x = 1 \). Therefore,
That derivation is the essence of what is the derivative of tangent. Notice how the identity simplifies the expression cleanly.
Alternative Proof Using the Limit Definition
You can also prove the derivative using the limit definition of the derivative: \( \lim_{h \to 0} \frac{\tan(x+h) – \tan x}{h} \). Rewrite tan(x+h) = sin(x+h)/cos(x+h), combine fractions, and apply the trig limits \( \lim_{h\to0} \sin h / h = 1 \) and \( \lim_{h\to0} (1-\cos h)/h = 0 \). After algebra, you’ll arrive at sec²(x). Many textbooks present this approach to reinforce limit skills. For a detailed walkthrough, see Khan Academy’s proof.
When the Argument Is a Function: The Chain Rule
The identity above is for the simple case where the argument is x. But many problems involve composite functions like tan(2x), tan(x²+1), or even tan(ln x). In those situations, you need the chain rule. The generalized answer captures what is the derivative of tangent when the tangent contains a nested function.
This is the generalized form. The “inner” derivative multiplies the outer derivative.
Worked Example 1: Derivative of tan(3x + 2)
🧪 Worked example
Find the derivative of tan(3x + 2).
Step 1: Outer function: tan(u) with u = 3x + 2.
Step 2: Outer derivative: d/du tan(u) = sec²(u).
Step 3: Inner derivative: du/dx = 3.
Step 4: Multiply: d/dx tan(3x+2) = sec²(3x+2) · 3.
Answer: \(3 \sec^2(3x+2)\).
Worked Example 2: Derivative of tan(e^x)
🧪 Worked example
Find the derivative of tan(e^x).
Step 1: Outer: tan(u) with u = e^x.
Step 2: Outer derivative: sec²(u).
Step 3: Inner derivative: du/dx = e^x.
Step 4: Multiply: d/dx tan(e^x) = sec²(e^x) · e^x.
Answer: \(e^x \sec^2(e^x)\).
Both examples demonstrate what is the derivative of tangent in action with a coefficient or exponential inside. Without the chain rule, you would miss the inner factor entirely.
Read on derivative of tan x, cos and sin
✅ When to use the chain rule
- Argument is a polynomial: tan(3x³ + x)
- Argument is a trig function: tan(sin x)
- Argument is an exponential: tan(e²ˣ)
- Argument is a log: tan(ln x)