Unit 05 · Lesson 06 · Week 14 · Differentiation Rules & Higher Derivatives
Derivatives of Exponential Functions
Exponential derivatives preserve exponential structure. The natural exponential function is its own derivative; other bases introduce a logarithmic scale factor. This lesson emphasizes derivatives of exponential functions through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in derivatives of exponential functions.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do derivative rules make complex rates of change efficient, interpretable, and verifiable?
Visual model
Exponential derivatives preserve exponential structure. The natural exponential function is its own derivative; other bases introduce a logarithmic scale factor.
(e^x)'=e^x(a^x)'=a^x\ln a
Worked reasoning
Misconception check: The derivative of a^x is not x a^{x-1}; that is the power rule for x^a.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Derivatives of Exponential Functions
Student Worksheet · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Differentiate e^x.
Differentiate 4^x.
Differentiate e^(3x).
Find the tangent slope of e^x at x=0.
Differentiate 5e^(−2x).
Distinguish d/dx(x³) from d/dx(3^x).
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.