Calculus 1 Equation Sheet
Derivatives
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\[ f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h} \]
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\[ \frac{d}{dx}(uv)=uv'+u'v \]
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\[ \frac{d}{dx}\left(\frac{t}{b}\right)=\frac{bt'-b't}{b^2} \]
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\[ \frac{d}{dx}f(u)=f'(u)u' \]
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\[ \frac{d}{dx}x^n=nx^{n-1} \]
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\[ \frac{d}{dx}\ln(x)=\frac{1}{x} \]
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\[ \frac{d}{dx}\log_a(x)=\frac{1}{x\ln(a)} \]
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\[ \frac{d}{dx}e^{ax}=ae^{ax} \]
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\[ \frac{d}{dx}a^x=a^x\ln(a) \]
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\[ \frac{d}{dx}\sin(x)=cos(x) \]
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\[ \frac{d}{dx}\cos(x)=-\sin(x) \]
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\[ \frac{d}{dx}tan(x)=\sec^2(x) \]
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\[ \frac{d}{dx}\cot(x)=-\csc^2(x) \]
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\[ \frac{d}{dx}\sec(x)=\sec(x)\tan(x) \]
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\[ \frac{d}{dx}\csc(x)=-\csc(x)\cot(x) \]
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\[ \frac{d}{dx}\arcsin(x)=\frac{1}{\sqrt{1-x^2}} \]
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\[ \frac{d}{dx}\arctan(x)=\frac{1}{1+x^2} \]
Integrals
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\[ \int x^ndx=\frac{x^{n+1}}{n+1}+C, n\neq 1 \]
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\[ \int\frac{1}{x}dx=\ln(x)+C \]
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\[ \int e^{ax}dx=\frac{1}{a}e^{ax}+C \]
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\[ \int a^xdx=\frac{a^x}{\ln(a)}+C \]
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\[ \int\cos(x)dx=\sin(x)+C \]
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\[ \int\sin(x)dx=-\cos(x)+C \]
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\[ \int\sec^2(x)dx=\tan(x)+C \]
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\[ \int\tan(x)dx=-\ln|\cos(x)|+C \]
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\[ \int\cot(x)dx=\ln|\sin(x)|+C \]
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\[ \int\sec(x)\tan(x)dx=\sec(x)+C \]
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\[ \int\csc^2(x)dx=-\cot(x)+C \]
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\[ \int\csc(x)\cot(x)dx=-\csc(x)+C \]
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\[ \int\frac{u'}{a^2+u^2}du=\frac{1}{a}\arctan\left(\frac{u}{a}\right)+C \]
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\[ \int\frac{u'}{\sqrt{a^2-u^2}}du=\arcsin\left(\frac{u}{a}\right)+C \]
Second Fundamental Theorem
\[
\frac{d}{dx}\int_u^vf(t)dt=f(v)v'-f(u)u'
\]
Approximations
\[
A=\frac{1}{2}w(h_1+h_2)\tag{Area of Trapezoid}
\]
\[
A\approx\frac{1}{2}w(h_1+2h_2+\ldots+2h_n+h_{n+1})\tag{Trapezoidal Rule}
\]
\[
A=\sum_{i=1}^n\Delta x f(x_i)\tag{Riemann Sum}
\]
Rate of Change
\[
\frac{f(b)-f(a)}{b-a}\tag{Average Rate of Change}
\]
\[
f'(c)\tag{Instantaneous Rate of Change}
\]
\[
f'(c)=\frac{f(b)-f(a)}{b-a}\tag{Mean Value Theorem}
\]
\[
f_{avg}=\frac{\int_a^bf(x)dx}{b-a}\tag{Average Value of a Function}
\]
\[
f(a)< K