Find derivatives of logarithmic functions

Learning Objectives State the derivative of the natural logarithm function, f(x) = ln(x). Apply the Chain Rule to find the derivative of composite logarithmic functions, such as f(x) = ln(u(x)). Derive and apply the formula for the derivative of a logarithmic function with any base, f(x) = log_b(x). Combine the Chain Rule with the general logarithmic derivative rule for functions of the form f(x) = log_b(u(x)). Use the properties of logarithms (product, quotient, power rules) to simplify complex functions before differentiating. Apply the technique of logarithmic differentiation to find derivatives of functions with variables in the exponent or complex products/quotients. How can we find the instantaneous rate of change for phenomena measured on a logarithmic scale, like ear...

TermDefinitionExample Natural Logarithm (ln)A logarithm with a special base called 'e', where e is an irrational number approximately equal to 2.71828. The function f(x) = ln(x) is the inverse of the exponential function g(x) = e^x.ln(e^2) = 2 Logarithmic FunctionA function of the form f(x) = log_b(x), where 'b' is a positive constant called the base (b ≠ 1). It answers the question: 'what exponent do we need for base b to get x?'log_10(100) = 2, because 10^2 = 100. Change of Base FormulaA formula used to convert a logarithm from one base to another, which is essential for deriving the general differentiation rule. The formula is log_b(a) = ln(a) / ln(b).log_3(7) = ln(7) / ln(3) Chain Rule with LogarithmsAn application of the Chain Rule for a composite functi...

Derivative of the Natural Logarithm \frac{d}{dx}[\ln(x)] = \frac{1}{x}, \text{ for } x > 0 This is the fundamental rule for the derivative of the natural log function. It states that the slope of the tangent line to the graph of ln(x) at any point x is simply 1/x. Chain Rule with the Natural Logarithm \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} = \frac{u'}{u} Use this rule when the argument of the natural logarithm is a function of x (let's call it u). The derivative is the derivative of the inside function (u') divided by the original inside function (u). Derivative of a General Logarithm \frac{d}{dx}[\log_b(x)] = \frac{1}{x \ln(b)} This rule is for logarithms with any base 'b'. It is similar to the natural log rule, but includ...