This concept distinguishes power functions from exponential functions and establishes their respective differentiation rules: the power rule (d/dx[xⁿ] = n·x^(n−1)) applies only when the variable is the base, while the exponential derivative rule (d/dx[aˣ] = ln(a)·aˣ) applies when the variable is the exponent, and combines these with the chain rule for differentiating compositions involving natural logarithms and exponentials. It also relies on the inverse relationship between the natural logarithm and the exponential function (ln(eˣ) = x) as a simplification tool prior to differentiation. This belongs to single-variable calculus, within the theory of differentiation rules for transcendental functions, building on the chain rule and logarithm properties.
This concept distinguishes power functions from exponential functions and establishes their respective differentiation rules: the power rule (d/dx[xⁿ] = n·x^(n−1)) applies only when the variable is t…