What is special about eˣ and ln x?
You will be able to: Use natural exponential and logarithm derivatives with correct domains.
What is special about eˣ and ln x?
An idealized quantity modeled by eˣ has a local rate numerically equal to its current amount. Its inverse, ln x, grows more slowly as x becomes larger.
A useful starting point: How are sine and cosine slopes related? →
Words and symbols before equations
- e
- The positive constant approximately 2.71828, the base of the natural exponential.
- Natural logarithm
- ln x, the inverse of eˣ, defined for x>0.
- Inverse relationship
- ln(eˣ)=x and e^(ln x)=x for x>0.
- Natural exponential
- e raised to a variable input, distinct from a power with fixed exponent.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Positive sample inputs permit either choice. Exponential has full real domain; ln and its derivative require x>0. e≈2.71828.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- exp at input 1: value=2.71828, derivative=2.71828. Blue: function; orange: tangent; second graph: derivative. Curves outside the finite vertical scale are clipped, not capped.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
The derivative of eˣ is eˣ. In its difference quotient, factoring eˣ leaves eˣ(eʰ−1)/h; the characteristic limit of the final factor is 1.
The derivative of ln x is 1/x for x>0. It has positive slope throughout its domain, with smaller slopes at larger positive inputs.
Do not apply the fixed-exponent power rule to eˣ: the base is constant and the exponent varies. Conversely x³ has variable base and fixed exponent.
Unit 2 applies these standard rules to uncomposed functions and linear combinations. General logarithm differentiation and chain-rule compositions are developed later.
A worked example, step by step
Find f′(1) for f(x)=2eˣ+3ln x.
- The original real domain is x>0.
- Differentiate term by term: f′(x)=2eˣ+3/x.
- Substitute x=1 to obtain 2e+3.
- This is approximately 8.437; retain the exact form 2e+3 when requested.
ln x is not defined for nonpositive real x, and the power rule does not apply to eˣ as though x were a constant exponent.
Is the derivative of ln x equal to 1/ln x?
Compare with an explanation
No. It is 1/x; at x=1 its value is 1 although ln 1=0.
Predict. Change one thing. Explain.
Switch from the exponential to logarithm. Compare the function value and its slope at several valid inputs; identify where their numbers happen to agree.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
exp at input 1: value=2.71828, derivative=2.71828. Blue: function; orange: tangent; second graph: derivative. Curves outside the finite vertical scale are clipped, not capped.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Positive sample inputs permit either choice. Exponential has full real domain; ln and its derivative require x>0. e≈2.71828.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor g(x)=eˣ−2ln x+x², find g′(x), its domain and g′(1).
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Original and derivative domains here are x>0.
- 1 point: Derivative of eˣ is eˣ and of −2ln x is −2/x.
- 1 point: Add derivative 2x to obtain g′=eˣ−2/x+2x.
- 1 point: At 1 the derivative is e−2+2=e.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Which function equals its own derivative?
eˣ.
RECALL 2Natural-log derivative and domain?
1/x for x>0.
RECALL 3Why not use the power rule on eˣ?
The exponent varies while the base is fixed.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What is special about eˣ and ln x?
- (eˣ)′=eˣ for all real x.
- (ln x)′=1/x for x>0.
Remember: ln x is not defined for nonpositive real x, and the power rule does not apply to eˣ as though x were a constant exponent.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Positive sample inputs permit either choice. Exponential has full real domain; ln and its derivative require x>0. e≈2.71828.
Refresh Kid · AP Calculus AB Unit 2 · Objectives FUN-3.A, LIM-3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.7, FUN-3.A, LIM-3.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.1, 3.3 and 3.5 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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