Calculus Reference Sheet

Derivative Rules:

Power Rule

• \(\frac{d}{dx}(x^n) = nx^{n-1}\) where \(n\) is any real number.

\(e^x\) Rule

• \(\frac{d}{dx}(e^x) = e^x\)

Natural Logarithm Rule

• \(\frac{d}{dx}(\ln(x)) = \frac{1}{x}\) where \(x > 0\)

Exponential Rule

• \(\frac{d}{dx}(b^x) = b^x\ln(b)\) where \(b > 0\) and \(b \neq 1\)

Logarithm Rule

• \(\frac{d}{dx}(\log_b(x)) = \frac{1}{x\ln(b)}\) where \(x > 0\), \(b > 0\), and \(b \neq 1\)

Constant Multiple Rule

• \(\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))\) where \(c\) is a real number.

Sum and Difference Rule

• \(\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))\)
  • \(\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))\)
  • \(\frac{d}{dx}(f(x) - g(x)) = \frac{d}{dx}(f(x)) - \frac{d}{dx}(g(x))\)

Product Rule

• \(\frac{d}{dx}(f(x)g(x)) = (\frac{df}{dx}(x))g(x) + f(x)\frac{dg}{dx}(x)\)

Quotient Rule

• \(\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{(\frac{df}{dx}(x))g(x) - f(x)\frac{dg}{dx}(x)}{(g(x))^2}\)

Chain Rule

• \(\frac{d}{dx}(f(g(x))) = \frac{df}{dg}\frac{dg}{dx}\)
  • Given \(h(x) = f(g(x))\), \(h'(x) = f'(g(x))g'(x)\).

Definition of Logarithm:

• For \(x>0\), \(b>0\), and \(b\neq 1\), \(y = \log_b(x)\) is equivalent to \(b^y=x\)

Properties of Logarithms:

Product Property

• \(\log_b(ac) = \log_b(a) + \log_b(c)\)

Quotient Property

• \(\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\)

Power Property

• \(\log_b(a^n) = n\log_b(a)\)

Properties of Sums:

• \(\sum_{i=1}^{n}c = n c\)

• \(\sum_{i=1}^{n}c a_i = c\sum_{i=1}^{n}a_i\)

• \(\sum_{i=1}^{n}a_i \pm b_i = \sum_{i=1}^{n}a_i \pm \sum_{i=1}^{n}b_i\)

  • \(\sum_{i=1}^{n}a_i + b_i = \sum_{i=1}^{n}a_i + \sum_{i=1}^{n}b_i\)
  • \(\sum_{i=1}^{n}a_i - b_i = \sum_{i=1}^{n}a_i - \sum_{i=1}^{n}b_i\)