The derivative of arctan(x) is 1/(1+x²). Derived via implicit differentiation of tan(y)=x and using the identity sec²(y)=1+tan²(y).

Essential Guide: What Is the Derivative of Arctan? (In 3 Steps)

If you’ve ever asked what is the derivative of arctan, you’re not alone. This formula is a cornerstone of calculus, especially when working with inverse trigonometric functions. Whether you’re preparing for an exam or refreshing your differentiation skills, here’s a clean, step‑by‑step explanation that starts from first principles and ends with practical examples.

✅ Quick answer: The derivative of arctan(x) is $$ \frac{d}{dx}\arctan(x)=\frac{1}{1+x^2}, \quad \text{for all real } x. $$ If you need the derivative with a chain rule — say, $\arctan(u)$ — it’s $\frac{u’}{1+u^2}$.

🔑 Key Takeaways

The derivative of arctan(x) is $1/(1+x^2)$.
Derivation uses implicit differentiation on $y=\arctan(x)$ (or $x=\tan(y)$).
The identity $\sec^2(y)=1+\tan^2(y)$ is essential.
Chain rule: derivative of $\arctan(u)$ is $u’/(1+u^2)$.
This formula is defined for all real numbers — no domain restrictions.

What Is the Derivative of Arctan? The Formula

The fundamental answer to what is the derivative of arctan is surprisingly simple: $$ \frac{d}{dx}\bigl(\arctan x\bigr)=\frac{1}{1+x^2}. $$ Unlike the derivatives of sine and cosine, which are cyclic, the derivative of arctan is a rational function. It is always positive, never zero, and its graph is a bell‑shaped curve that peaks at $x=0$ (where the slope is $1$).

But what is the derivative of arctan when the argument is a function of $x$? If $u(x)$ is differentiable, then $$ \frac{d}{dx}\bigl(\arctan u\bigr)=\frac{u’(x)}{1+[u(x)]^2}. $$ This chain‑rule extension is crucial for almost every real‑world application.

Deriving the Derivative of Arctan Step by Step

To truly understand what is the derivative of arctan, it helps to derive it yourself. Follow these three steps:

Set up the inverse relation
Let $y = \arctan x$. Then $\tan y = x$. This is the starting point for implicit differentiation.

Differentiate implicitly
Differentiate both sides with respect to $x$: $$ \sec^2 y \cdot \frac{dy}{dx} = 1. $$

Use the Pythagorean identity
Recall $\sec^2 y = 1 + \tan^2 y = 1 + x^2$. Substitute: $$ (1+x^2)\frac{dy}{dx} = 1 \quad\Rightarrow\quad \frac{dy}{dx} = \frac{1}{1+x^2}. $$ Done!

That’s it. The derivation elegantly answers what is the derivative of arctan in a few lines. In practice, many students memorise the formula, but understanding the steps helps prevent mistakes.

💡 Pro tip: When you differentiate $\arctan(\text{something})$, always check if you need the chain rule. For example, $\frac{d}{dx}\arctan(3x) = \frac{3}{1+(3x)^2} = \frac{3}{1+9x^2}$.

Why the Focus on “What Is the Derivative of Arctan?” Matters

Understanding what is the derivative of arctan is not just an academic exercise. This derivative appears in:

Integration – The antiderivative $\int \frac{1}{1+x^2}\,dx = \arctan x + C$ is a standard result.
Physics – In circuits involving RC time constants and in certain applications of inverse functions.
Optimization – When finding angles that maximize or minimize a quantity, arctan derivatives often show up.

But the question what is the derivative of arctan also naturally leads to questions about its siblings: for instance, what is the derivative of tanx or what is the derivative of secx. These are covered in our comprehensive guide on derivatives of a trig function.

Relationship to Other Inverse Trig Derivatives

The derivative of arcsin(x) is $1/\sqrt{1-x^2}$, and the derivative of arccos(x) is $-1/\sqrt{1-x^2}$. Compared with these, what is the derivative of arctan stands out because it has no square root and is defined for all real $x$. That simplicity makes it a favourite in calculus problems.

🎯 From experience: I often see students confuse the derivative of arctan with the derivative of arcsin. A quick memory aid: arctan involves a sum ($1+x^2$), while arcsin involves a difference ($1-x^2$).

Worked Examples: Putting the Derivative of Arctan to Use

Let’s apply the answer to what is the derivative of arctan in a few concrete problems.

🧪 Worked example 1

Find $f’(x)$ if $f(x)=\arctan(5x)$.
Solution: Use the chain rule with $u=5x$, $u’=5$. Then $$ f’(x) = \frac{5}{1+(5x)^2} = \frac{5}{1+25x^2}. $$

🧪 Worked example 2

Differentiate $g(x)=x^2\arctan(x)$.
Solution: Apply product rule: $$ g’(x) = 2x\arctan(x) + x^2\cdot\frac{1}{1+x^2} = 2x\arctan(x) + \frac{x^2}{1+x^2}. $$

🧪 Worked example 3

Find $\frac{d}{dx}\bigl(\arctan(\sqrt{x})\bigr)$.
Solution: Let $u=\sqrt{x}=x^{1/2}$, then $u’=\frac{1}{2\sqrt{x}}$. So $$ \frac{d}{dx} = \frac{\frac{1}{2\sqrt{x}}}{1+(\sqrt{x})^2} = \frac{1}{2\sqrt{x}\,(1+x)}. $$

Every example reinforces what is the derivative of arctan and how the chain rule extends it. In practice, you’ll rarely differentiate just $\arctan$ alone — the argument is almost always a function.

Common Mistakes When Finding the Derivative of Arctan

Even after learning what is the derivative of arctan, students slip up. Here are the top three errors:

⚠️ Avoid this: Forgetting the chain rule is the #1 mistake. If you differentiate $\arctan(x^2)$ and write $\frac{1}{1+x^2}$ instead of $\frac{2x}{1+x^4}$, you’ll lose points. Always differentiate the inner function!

⚠️ Avoid this: Mixing up the derivative of arctan with arcsin or arccot. The derivative of arctan is $1/(1+x^2)$; the derivative of arccot is $-1/(1+x^2)$. Watch the sign!

⚠️ Avoid this: Thinking the derivative of arctan is $\frac{1}{1-x^2}$ (that’s the derivative of arctanh, the hyperbolic arctan). The hyperbolic and circular functions are different — keep them separate.

Visualising the Derivative of Arctan

A picture often cements what is the derivative of arctan. The graph of $y=\arctan x$ is a gentle S‑curve approaching horizontal asymptotes at $\pm\frac{\pi}{2}$. Its slope — the derivative $1/(1+x^2)$ — is highest at $x=0$ and tapers off as $x$ moves away from zero.

If you prefer a narrated walkthrough, the video above shows the exact steps. It’s a great supplement to the written explanation of what is the derivative of arctan.

Connecting to the Derivative of Tan and Other Trig Functions

Once you master what is the derivative of arctan, the next logical step is exploring its “parent” function. The derivative of tan(x) is $\sec^2(x)$ — notice that this is exactly the reciprocal of $1/(1+x^2)$ when you replace $x$ with $\tan y$. This relationship is why the implicit derivation works so neatly. For a deep dive, see our article on what is the derivative of tangent.

Similarly, the derivative of arcsec and arccsc follow analogous patterns. If you’re curious about what is the derivative of sec and its inverse, we cover that too. For the basic trig derivatives, our sibling article on what is the derivative of sin gives a solid foundation.

All these live under the umbrella of derivatives of a trig function, our hub that organises every formula and proof.

Table: Quick Reference for Inverse Trig Derivatives

Function	Derivative	Domain
$\arcsin x$	$1/\sqrt{1-x^2}$	$-1
$\arccos x$	$-1/\sqrt{1-x^2}$	$-1
$\arctan x$	$1/(1+x^2)$	all real $x$
$\mathrm{arccot}\,x$	$-1/(1+x^2)$	all real $x$

This table highlights why what is the derivative of arctan is the cleanest of the bunch — no square roots, no domain restrictions, just a rational function.

Why You’ll Keep Asking “What Is the Derivative of Arctan?”

The question what is the derivative of arctan pops up repeatedly in calculus II, differential equations, and even in engineering contexts. Because arctan appears in integrals like $\int \frac{dx}{a^2+x^2} = \frac{1}{a}\arctan(x/a) + C$, knowing the derivative inside out helps you spot patterns quickly.

In summary, what is the derivative of arctan is $\frac{1}{1+x^2}$, derived by implicit differentiation of $\tan y = x$. With the chain rule, the formula becomes $\frac{u’}{1+u^2}$. That’s all you need to solve a huge variety of problems.

🤔 Did you know? The derivative of arctan(x) is directly related to the probability density function of the Cauchy distribution. Yes — the same $1/(1+x^2)$ formula appears in statistics!

Frequently Asked Questions

What is the derivative of arctan(x)?+

The derivative of arctan(x) is 1/(1+x²). This formula applies for all real x.

How do you derive the derivative of arctan?+

Start with y = arctan(x), so tan(y)=x. Implicitly differentiate: sec²(y) dy/dx = 1. Use identity sec²(y)=1+tan²(y)=1+x², then solve for dy/dx = 1/(1+x²).

Why is the derivative of arctan always positive?+

Because the denominator 1+x² is always positive, so 1/(1+x²) > 0 for all x. The arctan function is strictly increasing, confirming the positive slope.

What are the common mistakes when finding the derivative of arctan?+

A common mistake is forgetting the chain rule when the argument is not x. For arctan(u), the derivative is u’/(1+u²). Also, confusing the positive sign is rare, but forgetting the denominator is a frequent error.

What is the integral of the derivative of arctan?+

The integral of 1/(1+x²) dx is arctan(x) + C, which is the inverse relation confirming the derivative.

Ready to go further?

Practice with more examples of inverse trig derivatives.

Explore the hub: Derivatives of a Trig Function →

External references: For a comprehensive overview of inverse trigonometric functions, see Wikipedia’s page. Another excellent resource is Paul’s Online Math Notes on derivatives of inverse trig functions.