Calculus and Exponential Rules: Derivatives, Logarithms, and Applications

Product Rule d/dx [f(x)g(x)]

f′(x)g(x) + g′(x)f(x)

Quotient Ruled/dx [f(x)/g(x)]

[f′(x)g(x) − g′(x)f(x)] / [g(x)]²

Chain Rule d/dx [f(g(x))]

f′(g(x)) · g′(x)

Exponent Rule: ax·ay

a^(x + y)

Exponent Rule: ax/ay

a^(x − y)

Exponent Rule: 1/ay

a^(−y)

Exponent Rule: (ab)^x

a^x · b^x

Exponent Rule: (a/b)^x

a^x / b^x

Exponent Rule: a^(xy)

(a^x)^y

Exponent Rule: a⁰

1

Exponent Rule: a¹

a

Log Rule: ln(ab)

ln(a) + ln(b)

Log Rule: ln(a/b)

ln(a) − ln(b)

Log Rule: ln(1/b)

−ln(b)

Log Rule: ln(a^x)

x · ln(a)

Log Rule: ln(1)

0

Log Rule: ln(e)

1

Derivative of e^x d/dx (e^x)

e^x

Derivative of ln(x)

1/x

To find maxima or minima

Take the derivative, set it equal to 0, and solve for x.

Exponential Model

C(t) = C₀e^(±kt), where + is for growth and − is for decay.

Half-Life Formula

t₁/₂ = −ln(1/2) / k