Calculus 1 Equation Sheet
Derivatives
\[
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}
\]
\[
\frac{d}{dx}(uv)=uv'+u'v
\]
\[
\frac{d}{dx}\left(\frac{t}{b}\right)=\frac{bt'-b't}{b^2}
\]
\[
\frac{d}{dx}f(u)=f'(u)u'
\]
\[
\frac{d}{dx}x^n=nx^{n-1}
\]
\[
\frac{d}{dx}\ln(x)=\frac{1}{x}
\]
\[
\frac{d}{dx}\log_a(x)=\frac{1}{x\ln(a)}
\]
\[
\frac{d}{dx}e^{ax}=ae^{ax}
\]
\[
\frac{d}{dx}a^x=a^x\ln(a)
\]
\[
\frac{d}{dx}\sin(x)=cos(x)
\]
\[
\frac{d}{dx}\cos(x)=-\sin(x)
\]
\[
\frac{d}{dx}tan(x)=\sec^2(x)
\]
\[
\frac{d}{dx}\cot(x)=-\csc^2(x)
\]
\[
\frac{d}{dx}\sec(x)=\sec(x)\tan(x)
\]
\[
\frac{d}{dx}\csc(x)=-\csc(x)\cot(x)
\]
\[
\frac{d}{dx}\arcsin(x)=\frac{1}{\sqrt{1-x^2}}
\]
\[
\frac{d}{dx}\arctan(x)=\frac{1}{1+x^2}
\]
Integrals
\[
\int x^ndx=\frac{x^{n+1}}{n+1}+C, n\neq -1
\]
\[
\int\frac{1}{x}dx=\ln(x)+C
\]
\[
\int e^{ax}dx=\frac{1}{a}e^{ax}+C
\]
\[
\int a^xdx=\frac{a^x}{\ln(a)}+C
\]
\[
\int\cos(x)dx=\sin(x)+C
\]
\[
\int\sin(x)dx=-\cos(x)+C
\]
\[
\int\sec^2(x)dx=\tan(x)+C
\]
\[
\int\tan(x)dx=-\ln|\cos(x)|+C
\]
\[
\int\cot(x)dx=\ln|\sin(x)|+C
\]
\[
\int\sec(x)\tan(x)dx=\sec(x)+C
\]
\[
\int\csc^2(x)dx=-\cot(x)+C
\]
\[
\int\csc(x)\cot(x)dx=-\csc(x)+C
\]
\[
\int\frac{du}{a^2+u^2}=\frac{1}{a}\arctan\left(\frac{u}{a}\right)+C
\]
\[
\int\frac{du}{\sqrt{a^2-u^2}}=\arcsin\left(\frac{u}{a}\right)+C
\]
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
\[
\frac{f(b)-f(a)}{b-a}\tag{Average Rate of Change}
\]
\[
f'(c)\tag{Instantaneous Rate of Change}
\]
\[
f'(c)=\frac{f(b)-f(a)}{b-a}\tag{Mean Value Theorem}
\]
\[
f_{avg}=\frac{\int_a^bf(x)dx}{b-a}\tag{Average Value of a Function}
\]
\[
f(a) < K < f(b), a < c < b,f(c)=K\tag{Intermediate Value Theorem}
\]