Derivative and Algebraic Rules

Constant

Derivative is 0, e.g., f(x) = 5, f'(x) = 0

Power Rule

Derivative is nx^(n-1), e.g., f(x) = 3x^2, f'(x) = 6x

Sum Rule

Derivative is f'(x) + g'(x), e.g., f(x) = 2x + 3, g(x) = x^2 - 1, (f+g)'(x) = 2 + 2x

Constant Multiple Rule

Derivative is cf'(x), e.g., f(x) = 4x^3, f'(x) = 12x^2, 3f'(x) = 36x^2

Product Rule

Derivative is f'(x)g(x) + f(x)g'(x), e.g., f(x) = x^2, g(x) = 2x + 1, (f * g)'(x) = 2x(2x+1) + x^2

Quotient Rule

Derivative is (f'(x)g(x) - f(x)g'(x))/(g(x))^2, e.g., f(x) = 2x, g(x) = x^2, (f/g)'(x) = (2x * x^2 - 2x^2)/(x^2)^2

Chain Rule

Derivative is f'(g(x)) g'(x), e.g., f(x) = √x, g(x) = x^2, (f ∘ g)'(x) = 1/(2√x) 2x

Exponential Rule

Derivative is a^x ln(a), e.g., f(x) = 2^x, f'(x) = 2^x ln(2)

Natural Logarithm Rule

Derivative is 1/x, e.g., f(x) = ln(x), f'(x) = 1/x

General Logarithmic Rule

Derivative is 1/(xln(a)), e.g., f(x) = log_a(x), f'(x) = 1/(xln(a))

Exponent Rule

Derivative is a^x ln(a) n, e.g., f(x) = e^(2x), f'(x) = e^(2x) * 2

Logarithm Property Rule

Derivative is 1/(xln(b)), e.g., f(x) = log_b(x), f'(x) = 1/(xln(b))

a^m * a^n = a^(m+n)

e.g., 2^3 * 2^4 = 2^(3+4) = 2^7

a^m / a^n = a^(m-n)

e.g., 3^5 / 3^2 = 3^(5-2) = 3^3

(a^m)^n = a^(mn)

e.g., (4^2)^3 = 4^(2*3) = 4^6

a^-m = 1/a^m

e.g., 5^-3 = 1/5^3

a^0 = 1

e.g., 6^0 = 1

log_b(xy) = log_b(x) + log_b(y)

e.g., log_2(4 * 8) = log_2(4) + log_2(8)

log_b(x/y) = log_b(x) - log_b(y)

e.g., log_3(9/3) = log_3(9) - log_3(3)

log_b(x^m) = m * log_b(x)

e.g., log_4(2^5) = 5 * log_4(2)

log_b(1) = 0

e.g., log_5(1) = 0

log_b(b) = 1

e.g., log_6(6) = 1