Derivatives — Exponential & Logarithmic explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
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Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
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Exponential and logarithmic functions grow and change in special ways.
Their derivatives show rates of growth and decay.
This topic is used in calculus, science, finance, biology, and data modelling.
The derivative of is .
The derivative of ln x is for .
For a function inside another function, use the chain rule.
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Logarithmic derivatives often create a fraction.
Meaning: The rate of change of is itself.
Definition: .
This makes unique and important in calculus.
Meaning: Use chain rule when the exponent is a function.
Definition: .
Differentiate the exponent and multiply.
Meaning: Natural log differentiates to a reciprocal.
Definition: .
The input must be positive for ln x in reaalued school calculus.
Meaning: Use chain rule with logarithms.
Definition: '/u.
Differentiate the inside, then divide by the inside.
Easy
Question: Differentiate .
Medium
Question: Differentiate .
Exam-level
Question: Differentiate ln(2).
Forgetting chain rule
Why it happens: Only the outer function is differentiated.
Correct approach: Multiply by the derivative of the inside function.
Using for ln(2)
Why it happens: The inside function is ignored.
Correct approach: Use u'/u.
Treating every logarithm as ln
Why it happens: The base is not checked.
Correct approach: Check whether the log is natural log or another base.
Why is special?
Its derivative is the same function.
What is the derivative of ln u?
It is u' divided by u.
Do I always need chain rule?
Use chain rule when the input is more than just x.
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