Calculus and Trigonometry Reference Rules

A complete set of vocabulary flashcards covering basic differentiation rules, integration formulas, and trigonometric identities from the lecture reference sheets.

Product Rule for Differentiation

$$\frac{d}{dx}[u v] = u v' + v u'$$

Quotient Rule for Differentiation

$$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v u' - u v'}{v^2}$$

Derivative of a Constant

$$\frac{d}{dx}[c] = 0$$

Power Rule for Differentiation

$$\frac{d}{dx}[u^n] = n u^{n - 1} u'$$

Derivative of $x$

$$\frac{d}{dx}[x] = 1$$

Derivative of the Natural Logarithm

$$\frac{d}{dx}[\ln(u)]=\frac{1}{u}$$

Derivative of $e^u$

$$\frac{d}{dx}[e^u] = e^u u'$$

Derivative of Logarithm Base $a$

$$\frac{d}{dx}[\log_a(u)] = \frac{u'}{(\ln(a)) u}$$

Derivative of Exponential Base $a$

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

Derivative of Sine

$$\frac{d}{dx}[\sin(u)] = (\cos(u)) u'$$

Derivative of Cosine

$$\frac{d}{dx}[\cos(u)] = -(\sin(u)) u'$$

Derivative of Tangent

$$\frac{d}{dx}[\tan(u)] = (\sec^2(u)) u'$$

Derivative of Cotangent

$$\frac{d}{dx}[\cot(u)] = -(\csc^2(u)) u'$$

Derivative of Secant

$$\frac{d}{dx}[\sec(u)] = (\sec(u) \tan(u)) u'$$

Derivative of Cosecant

$$\frac{d}{dx}[\csc(u)] = -(\csc(u) \cot(u)) u'$$

Derivative of Arcsine

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

Derivative of Arccosine

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

Derivative of Arctangent

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

Derivative of Arccotangent

$$\frac{d}{dx}[\operatorname{arccot}(u)] = \frac{-u'}{1 + u^2}$$

Derivative of Arcsecant

$$\frac{d}{dx}[\operatorname{arcsec}(u)] = \frac{u'}{|u| \sqrt{u^2 - 1}}$$

Derivative of Arccosecant

$$\frac{d}{dx}[\operatorname{arccsc}(u)] = \frac{-u'}{|u| \sqrt{u^2 - 1}}$$

Constant Multiple Integration Rule

$$\int k f(u)\,du = k \int f(u)\,du$$

Sum and Difference Integration Rule

$$\int [f(u) \pm g(u)]\,du = \int f(u)\,du \pm \int g(u)\,du$$

Integral of $du$

$$\int du = u + C$$

Power Rule for Integration

$$\int u^n\,du = \frac{u^{n + 1}}{n + 1} + C,\, n \neq -1$$

Integral of $\frac{1}{u}$

$$\int \frac{du}{u} = \ln(|u|) + C$$

Integral of $e^u$

$$\int e^u\,du = e^u + C$$

Integral of Exponential Base $a$

$$\int a^u\,du = \left(\frac{1}{\ln(a)}\right) a^u + C$$

Integral of Sine

$$\int \sin(u)\,du = -\cos(u) + C$$

Integral of Cosine

$$\int \cos(u)\,du = \sin(u) + C$$

Integral of Tangent

$$\int \tan(u)\,du = -\ln(|\cos(u)|) + C$$

Integral of Cotangent

$$\int \cot(u)\,du = \ln(|\sin(u)|) + C$$

Integral of Secant

$$\int \sec(u)\,du = \ln(|\sec(u) + \tan(u)|) + C$$

Integral of Cosecant

$$\int \csc(u)\,du = -\ln(|\csc(u) + \cot(u)|) + C$$

Integral of $\sec^2(u)$

$$\int \sec^2(u)\,du = \tan(u) + C$$

Integral of $\csc^2(u)$

$$\int \csc^2(u)\,du = -\cot(u) + C$$

Integral of $\sec(u) \tan(u)$

$$\int \sec(u) \tan(u)\,du = \sec(u) + C$$

Integral of $\csc(u) \cot(u)$

$$\int \csc(u) \cot(u)\,du = -\csc(u) + C$$

Integral Yielding Arcsine

$$\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin\left(\frac{u}{a}\right) + C$$

Integral Yielding Arctangent

$$\int \frac{du}{a^2 + u^2} = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C$$

Integral Yielding Arcsecant

$$\int \frac{du}{u \sqrt{u^2 - a^2}} = \frac{1}{a} \operatorname{arcsec}\left(\frac{|u|}{a}\right) + C$$

Pythagorean Identities

$\sin^2(x) + \cos^2(x) = 1$, $1 + \tan^2(x) = \sec^2(x)$, and $1 + \cot^2(x) = \csc^2(x)$

Quotient Identities

$\tan(x) = \frac{\sin(x)}{\cos(x)}$ and $\cot(x) = \frac{\cos(x)}{\sin(x)}$

Reciprocal Identities

$\sin(x) = \frac{1}{\csc(x)}$, $\cos(x) = \frac{1}{\sec(x)}$, and $\tan(x) = \frac{1}{\cot(x)}$

Sum and Difference Formula for Sine

$$\sin(u \pm v) = \sin(u) \cos(v) \pm \cos(u) \sin(v)$$

Sum and Difference Formula for Cosine

$$\cos(u \pm v) = \cos(u) \cos(v) \mp \sin(u) \sin(v)$$

Sum and Difference Formula for Tangent

$$\tan(u \pm v) = \frac{\tan(u) \pm \tan(v)}{1 \mp \tan(u) \tan(v)}$$

Double-Angle Formula for Sine

$$\sin(2u) = 2 \sin(u) \cos(u)$$

Double-Angle Formulas for Cosine

$$\cos(2u) = \cos^2(u) - \sin^2(u) = 2 \cos^2(u) - 1 = 1 - 2 \sin^2(u)$$

Double-Angle Formula for Tangent

$$\tan(2u) = \frac{2 \tan(u)}{1 - \tan^2(u)}$$

Power-Reducing Formula for Sine

$$\sin^2(u) = \frac{1 - \cos(2u)}{2}$$

Power-Reducing Formula for Cosine

$$\cos^2(u) = \frac{1 + \cos(2u)}{2}$$

Power-Reducing Formula for Tangent

$$\tan^2(u) = \frac{1 - \cos(2u)}{1 + \cos(2u)}$$

Even/Odd Identities

Even functions satisfy $\cos(-x) = \cos(x)$ and $\sec(-x) = \sec(x)$; odd functions satisfy $\sin(-x) = -\sin(x)$, $\csc(-x) = -\csc(x)$, $\tan(-x) = -\tan(x)$, and $\cot(-x) = -\cot(x)$

Cofunction Identities

$\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$, $\cos\left(\frac{\pi}{2} - x\right) = \sin(x)$, and $\tan\left(\frac{\pi}{2} - x\right) = \cot(x)$