Chapter 3 Rules: Exponential, Logarithmic, Inverse Trigonometric, and Hyperbolic Functions

Vocabulary practice flashcards covering Chapter 3 formulas and definitions, including exponential and logarithmic rules, derivatives of inverse trigonometric functions, and definitions and derivatives of hyperbolic functions.

Product Rule of Logarithms

$$\log_a(xy) = \log_a(x) + \log_a(y)$$

Quotient Rule of Logarithms

$$\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$$

Power Rule of Logarithms

$$\log_a(x^y) = y\log_a(x)$$

Logarithm of 1 Rule

$$\log_a(1) = 0$$

Change of Base Formula

$$\log_a(x) = \frac{\ln(x)}{\ln(a)}$$

Derivative of $\log_a(x)$

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

Derivative of $a^x$

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

Derivative of $e^x$

$$\frac{d}{dx} e^x = e^x$$

Chain Rule for $a^u$

$$\frac{d}{dx} a^u = u' a^u \ln(a)$$

Chain Rule for $\ln(u)$

$$\frac{d}{dx} \ln(u) = \frac{u'}{u}$$

Chain Rule for $\log_b(u)$

$$\frac{d}{dx} \log_b(u) = \frac{u'}{u\ln(b)}$$

Logarithmic Differentiation

A procedure used when taking the logs of a complicated product or quotient: take the logarithms of both sides, differentiate implicitly, and then solve for $y'$.

Domain and Range of $\arcsin(x)$

Domain: $-1 \le x \le 1$; Range: $-\frac{\pi}{2} \le y \le \frac{\pi}{2}$

Derivative of $\arcsin(x)$

$$\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1 - x^2}}$$

Chain Rule for $\arcsin(u)$

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

Domain and Range of $\arccos(x)$

Domain: $-1 \le x \le 1$; Range: $0 \le y \le \pi$

Chain Rule for $\arccos(u)$

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

Domain and Range of $\arctan(x)$

Domain: $-\infty < x < \infty$; Range: $-\frac{\pi}{2} < y < \frac{\pi}{2}$

Derivative of $\arctan(x)$

$$\frac{d}{dx}\arctan(x) = \frac{1}{1 + x^2}$$

Chain Rule for $\arctan(u)$

$$\frac{d}{dx}\arctan(u) = \frac{u'}{1 + u^2}$$

Derivative of $\operatorname{arccsc}(x)$

$$\frac{d}{dx}\operatorname{arccsc}(x) = -\frac{1}{|x|\sqrt{x^2 - 1}}$$

Derivative of $\operatorname{arcsec}(x)$

$$\frac{d}{dx}\operatorname{arcsec}(x) = \frac{1}{|x|\sqrt{x^2 - 1}}$$

Hyperbolic Sine Function $\sinh(x)$

$$\sinh(x) = \frac{e^x - e^{-x}}{2}$$

Derivative of $\sinh(x)$

$$\frac{d}{dx}\sinh(x) = \cosh(x)$$

Derivative of $\sinh^{-1}(x)$

$$\frac{d}{dx}\sinh^{-1}(x) = \frac{1}{\sqrt{1 + x^2}}$$

Hyperbolic Cosine Function $\cosh(x)$

$$\cosh(x) = \frac{e^x + e^{-x}}{2}$$

Derivative of $\cosh(x)$

$$\frac{d}{dx}\cosh(x) = \sinh(x)$$

Derivative of $\cosh^{-1}(x)$

$$\frac{d}{dx}\cosh^{-1}(x) = \frac{1}{\sqrt{x^2 - 1}}$$

Hyperbolic Tangent Function $\tanh(x)$

$$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$$

Derivative of $\tanh(x)$

$$\frac{d}{dx}\tanh(x) = \operatorname{sech}^2(x)$$

Derivative of $\tanh^{-1}(x)$

$$\frac{d}{dx}\tanh^{-1}(x) = \frac{1}{1 - x^2}$$

Derivative of $\operatorname{csch}(x)$

$$\frac{d}{dx}\operatorname{csch}(x) = -\operatorname{csch}(x)\coth(x)$$

Derivative of $\operatorname{sech}(x)$

$$\frac{d}{dx}\operatorname{sech}(x) = -\operatorname{sech}(x)\tanh(x)$$

Derivative of $\coth(x)$

$$\frac{d}{dx}\coth(x) = -\operatorname{csch}^2(x)$$