Section 3.6: Derivatives of Inverse Functions

Flashcards covering Section 3.6: Derivatives of Inverse Functions, including the Inverse Function Derivative Rule, derivatives of logarithms, logarithmic differentiation, the generalized power rule, and derivatives of inverse trigonometric functions.

According to the Derivative Rule for Inverse Functions, what is the formula for $(f^{-1})'(f(x))$ when $x \in (a, b)$ and $f'(x) \neq 0$?

$$(f^{-1})'(f(x)) = \frac{1}{f'(x)}$$

According to the Derivative Rule for Inverse Functions, what is the formula for $(f^{-1})'(x)$ for $x \in f((a, b))$?

$$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$$

What is the derivative of the natural logarithm function $\ln(x)$ for all $x > 0$?

$$\frac{d}{dx}(\ln(x)) = \frac{1}{x}$$

Given a positive base $a \neq 1$, what is the derivative of $\log_a(x)$ for all $x > 0$?

$$\frac{d}{dx}(\log_a(x)) = \frac{1}{\ln(a)} \cdot \frac{1}{x}$$

What are the three steps of Logarithmic Differentiation?

Step 1: Apply the natural logarithm to both sides of $y = f(x)$ and simplify using log properties. Step 2: Assuming $y$ is differentiable, differentiate both sides implicitly with respect to $x$. Step 3: Solve the resulting equation for $y'$.

In the proof of the Power Rule using logarithmic differentiation for $y = x^r$, what intermediate step is taken before taking logarithms?

$|y| = |x|^r$, which allows taking the natural logarithm of both sides to get $\ln(|y|) = r \ln(|x|)$.

What is the Generalized Power Rule for a differentiable function $u = u(x)$?

$$\frac{d}{dx}(u^r) = r u^{r-1} \frac{du}{dx}$$

What are the domain and range of the inverse sine function $y = \sin^{-1}(x)$?

Domain: $[-1, 1]$; Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$

What are the domain and range of the inverse cosine function $y = \cos^{-1}(x)$?

Domain: $[-1, 1]$; Range: $[0, \pi]$

Which inverse trigonometric function is represented in this graph, and what are its stated domain and range?

The inverse tangent function $y = \tan^{-1}(x)$, with domain $(-\infty, \infty)$ and range $(-\frac{\pi}{2}, \frac{\pi}{2})$.

What is the identity that relates $\csc^{-1}(x)$ to $\sin^{-1}$?

$$\csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right)$$

What is the identity that relates $\sec^{-1}(x)$ to $\cos^{-1}$?

$$\sec^{-1}(x) = \cos^{-1}\left(\frac{1}{x}\right)$$

What identity relates $\cot^{-1}(x)$ to $\tan^{-1}$, and what is the value of $\cot^{-1}(0)$?

$\cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right)$, with $\cot^{-1}(0) = \frac{\pi}{2}$.

In the derivation of the derivative of $\tan^{-1}(x)$, what are the side lengths of this reference right triangle when $\theta = \tan^{-1}(x)$?

Opposite side = $x$, adjacent side = $1$, and hypotenuse = $\sqrt{1 + x^2}$.

What is the derivative of $\sin^{-1}(u)$ with respect to $x$ using the chain rule?

$$\frac{d}{dx}(\sin^{-1}(u)) = \frac{1}{\sqrt{1 - u^2}} \frac{du}{dx}$$

What is the derivative of $\cos^{-1}(u)$ with respect to $x$ using the chain rule?

$$\frac{d}{dx}(\cos^{-1}(u)) = -\frac{1}{\sqrt{1 - u^2}} \frac{du}{dx}$$

What is the derivative of $\tan^{-1}(u)$ with respect to $x$ using the chain rule?

$$\frac{d}{dx}(\tan^{-1}(u)) = \frac{1}{1 + u^2} \frac{du}{dx}$$

What is the derivative of $\cot^{-1}(u)$ with respect to $x$ using the chain rule?

$$\frac{d}{dx}(\cot^{-1}(u)) = -\frac{1}{1 + u^2} \frac{du}{dx}$$

What is the derivative of $\sec^{-1}(u)$ with respect to $x$ using the chain rule?

$$\frac{d}{dx}(\sec^{-1}(u)) = \frac{1}{|u|\sqrt{u^2 - 1}} \frac{du}{dx}$$

What is the derivative of $\csc^{-1}(u)$ with respect to $x$ using the chain rule?

$$\frac{d}{dx}(\csc^{-1}(u)) = -\frac{1}{|u|\sqrt{u^2 - 1}} \frac{du}{dx}$$