derivative of cos inverse
Understand the Problem
The question is asking for the derivative of the inverse cosine function, which involves calculus knowledge about derivatives and inverse trigonometric functions.
Answer
$-\frac{1}{\sqrt{1 - x^2}}$
Answer for screen readers
The derivative of $\arccos(x)$ is $-\frac{1}{\sqrt{1 - x^2}}$
Steps to Solve
- Identify the formula for the derivative of the inverse cosine function
The derivative of the inverse cosine function (also known as $\arccos(x)$) can be found using a known formula: $$ \frac{d}{dx} \arccos(x) = -\frac{1}{\sqrt{1 - x^2}} $$
- Understand the domain of the function
The function $\arccos(x)$ is defined for $-1 \leq x \leq 1$. This is important because the derivative also only applies within this domain.
The derivative of $\arccos(x)$ is $-\frac{1}{\sqrt{1 - x^2}}$
More Information
The inverse cosine function, $\arccos(x)$, is the angle whose cosine is $x$. The derivative of the inverse cosine shows how sensitive this angle is to changes in $x$ and is applicable within the domain $-1 \leq x \leq 1$.
Tips
A common mistake is to forget the negative sign in the derivative formula or to apply the formula outside of its domain.