Calculus I Equation Sheet

---

Basic Derivative Rules

Rule	Function Form	Derivative Form
Constant	\(c\)	\(0\)
Power	\(x^n\)	\(n x^{n-1}\)
Constant Multiple	\(c \cdot f(x)\)	\(c \cdot f'(x)\)
Sum	\(f(x) + g(x)\)	\(f'(x) + g'(x)\)
Difference	\(f(x) - g(x)\)	\(f'(x) - g'(x)\)
Product	\(f(x)g(x)\)	\(f'(x)g(x) + f(x)g'(x)\)
Quotient	\(\dfrac{f(x)}{g(x)}\)	\(\dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\)
Chain	\(f(g(x))\)	\(f'(g(x)) \cdot g'(x)\)
Inverse	\(f^{-1}(x)\)	\(\dfrac{1}{f'(f^{-1}(x))}\)

---

Trigonometric Derivatives

Function	Derivative
\(\sin x\)	\(\cos x\)
\(\cos x\)	\(-\sin x\)
\(\tan x\)	\(\sec^2 x\)
\(\sec x\)	\(\sec x \tan x\)
\(\csc x\)	\(-\csc x \cot x\)
\(\cot x\)	\(-\csc^2 x\)

---

Exponential & Logarithmic Derivatives

Function	Derivative
\(a^x\)	\(a^x \ln(a)\)
\(e^x\)	\(e^x\)
\(a^{g(x)}\)	\(g'(x)\, a^{g(x)} \ln(a)\)
\(e^{g(x)}\)	\(g'(x)\, e^{g(x)}\)
\(\log_a(x)\)	\(\dfrac{1}{x \ln(a)}\)
\(\ln(x)\)	\(\dfrac{1}{x}\)
\(\log_a(g(x))\)	\(\dfrac{g'(x)}{g(x)\ln(a)}\)
\(\ln(g(x))\)	\(\dfrac{g'(x)}{g(x)}\)

---

Average Rate of Change

\[ AV_[a,b] = \frac{f(b) - f(a)}{b-a} \]

Limit Definition of a Derivative

\[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]

---

General Functions & Forms

• Linear Forms:
  General: \(f(x) = ax+b\),
  Point-slope: \(y-y_1 = m(x-x_1)\),
  Slope-intercept: \(y=mx+b\)

• Quadratic Forms:
  General: \(ax^2+bx+c\),
  Vertex: \(a(x-h)^2+k\),
  Roots: \(\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)

• Polynomial (nth-order):
  \(f(x) = a_nx^n + \dots + a_1x + a_0\)

• Transformations:
  \(g(x) = a f(x-b) + c\)

• Sinusoid:
  \(f(x) = a \sin(kx+b)+c,\ \ T=\dfrac{2\pi}{k}\)

---

Applications

• Slope of tangent line: \(m = f'(a)\)
  
• Tangent line equation: \(y = f'(a)(x-a)+f(a)\)
  
• Linear approximation: \(L(x)=f(a)+f'(a)(x-a)\)
  
• Related rates: \(\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}\)
  
• Concavity: \(f''>0 \Rightarrow\) concave up, \(f''<0 \Rightarrow\) concave down
  
• Extrema: solve \(f'(x)=0\), check sign of \(f'\) or use \(f''\)

---