Derivatives of Inverse Trig Functions

An inverse trig function is simply the reverse of a trig function: you give it a ratio, and it gives you back the angle. ($\arcsin x$ means "the angle whose sine is $x$," and the "arc" in the name refers to the arc length on the unit circle, which is what an angle in radians measures.) They're how you turn measurements back into angles, like how far a camera has to turn. The formulas look odd, but they're just $\frac{1}{\sqrt{1 - x^2}}$ (the slope of $\arcsin$), the same thing with a minus sign (the slope of $\arccos$), and $\frac{1}{1 + x^2}$ (the slope of $\arctan$). Each one comes from a short, simple argument, so bear with it.

Deriving $\arcsin$. We don't have a formula to plug into the definition, but we do know how to differentiate sine. So turn the problem around. Let $y = \arcsin x$, which means $\sin y = x$ with $-\frac\pi2 \le y \le \frac\pi2$. Differentiate both sides with respect to $x$. The right side gives $1$. On the left, $y$ depends on $x$, so the derivative of $\sin y$ is $\cos y\cdot y'$ (you'll see why properly in the chain rule lesson). That gives $\cos y\cdot y' = 1$, so $y' = \frac{1}{\cos y}$.

That answer is in terms of $y$, and we want it in terms of $x$. We know $\sin y = x$, and $\sin^2 y + \cos^2 y = 1$, so $\cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - x^2}$. (We take the positive root because $\cos y \ge 0$ for $y$ in that range.) That's where the trig disappears to.

$\arctan$ works the same way, starting from $\tan y = x$: differentiating gives $\sec^2 y\cdot y' = 1$, and $\sec^2 y = 1 + \tan^2 y = 1 + x^2$.

Domains. $\arcsin$ and $\arccos$ are only differentiable for $-1 < x < 1$. You can see it in the formula: at $x = \pm1$ the square root in the denominator is zero, and the graph has a vertical tangent. $\arctan$ works for every $x$. Also notice $\arcsin x + \arccos x = \frac\pi2$, a constant, which is why their derivatives are negatives of each other.

Honestly, you'll use these mostly in reverse. In Calculus II, integrals like $\int\frac{dx}{1 + x^2} = \arctan x + C$ show up everywhere.

Inverse trig functions turn measurements back into angles: a camera's viewing angle, a robot joint's angle from its tool position, a ramp's slope angle. Their derivatives say how sensitive those angles are to measurement errors.