Mathematics

Definition of Logarithm:\(\text{If and only if } y = a^x ), where ( a > 0 ), ( a \neq 1 ), and ( x > 0 \)\(\log_a y = x \) Types of Log:1. Common log ( \(\log_{10} x \) )Also written as \(\log x \)2. Natural log ( \(\log_{2.71828} x \) )Also written…

Limit of a Constant:\( \lim \limits_{x \to a} c = c \) \( \lim \limits_{x \to a} x = a \)\(\lim \limits_{x \to a} \frac{x^n-a^n}{x-a} =na^{n-1} \)\(\lim \limits_{x \to a} \frac{x^m-a^m}{x^n-a^n} =\frac{m^{m-n}}{n} \)\(\lim \limits_{x \to 0} \frac{a^x-1}{x} = \ln a \) Exponential Functions:\( \lim \limits_{x \to a} e^x = e^a \)\( \lim \limits_{x \to 0}…

Constant Rule:\( \int k \, dx = k x + C \) Constant Multiple Rule:\( \int k \cdot f(x) \, dx = k \int f(x) \, dx\) Power Rule:\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1 \) \( \int (ax+b)^n \, dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C \) Exponential…

Power Rule: \( \frac{d}{dx} \left[ x^n \right] = n x^{n-1} \) Constant Rule: \( \frac{d}{dx} \left[ c \right] = 0 \) Constant Multiple Rule: \( \frac{d}{dx} \left[ k f(x) \right] = k \frac{d}{dx} f(x) \) Exponential Functions: \( \frac{d}{dx} \left[ e^x \right] = e^x \)\( \frac{d}{dx} \left[ e^{(ax+b)} \right] = ae^{(ax+b)} \)\( \frac{d}{dx} \left[ a^x…