Find derivatives of exponential functions

Learning Objectives State the derivative of the natural exponential function, f(x) = e^x. State the derivative of a general exponential function, f(x) = a^x. Apply the Chain Rule to find the derivatives of composite exponential functions like f(x) = e^u and f(x) = a^u, where u is a function of x. Combine the Product Rule with exponential differentiation rules to find derivatives. Combine the Quotient Rule with exponential differentiation rules to find derivatives. Solve problems involving the slope and equation of a tangent line to an exponential curve at a given point. Ever wondered how to calculate the exact rate of change of something growing incredibly fast, like a viral video's views or a savings account with compound interest? 📈 This tutorial focuses on the uniq...

TermDefinitionExample Exponential FunctionA function in the form f(x) = a^x, where the base 'a' is a positive constant not equal to 1, and the exponent 'x' is the variable.f(x) = 2^x or g(t) = 10^t The Number 'e' (Euler's Number)An irrational mathematical constant, approximately equal to 2.71828. It is the unique base for which the derivative of the exponential function a^x is the function itself.The value of e^1 is approximately 2.71828. Natural Exponential FunctionThe exponential function with base 'e', written as f(x) = e^x. Its derivative is uniquely itself, e^x.The function y = e^x models continuous growth. Natural Logarithm (ln)The logarithm to the base 'e'. It is the inverse of the natural exponential function, meaning ln(e^x)...

Derivative of the Natural Exponential Function d/dx(e^x) = e^x The simplest exponential derivative rule. The derivative of e^x is just e^x. This is a fundamental property of the number 'e'. Derivative of a General Exponential Function d/dx(a^x) = a^x * ln(a) Used for any exponential function with a positive base 'a' (where a ≠ 1). The derivative is the original function multiplied by the natural logarithm of its base. The Chain Rule for Exponential Functions d/dx(e^u) = e^u * du/dx and d/dx(a^u) = a^u * ln(a) * du/dx This is the most common and critical rule. Use it when the exponent is a function of x (represented by 'u'). You take the derivative of the outer exponential function and multiply it by the derivative of the inner function...