Difference quotient
The derivative is when this finite limit exists. Substitute everywhere in the original expression.
Calculus / English
Definition of the Derivative
The Organic Chemistry Tutor · YouTube · 23:31
YouTube · The Organic Chemistry TutorOriginal video
READ & KEEP
The explanation, unpacked.
Reviewed learning material · Video analysis · English01 02 03
Read the full overview
Work from the limit definition of the derivative, using cancellation and conjugates for polynomial, reciprocal and radical examples.
Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.
Chapters
0:00Difference quotient6:00Cancel before taking the limit12:00Rationalize a radicalLearning script
Generated from the video's visuals and explanation; not verbatim speech.
The derivative is when this finite limit exists. Substitute everywhere in the original expression.
For ², expanding the numerator gives 2xh+h². For nonzero h the quotient becomes , whose limit is 2x.
For =√x, multiply by the conjugate to cancel h. At the limit yields ; the formula does not give a finite derivative at .
Knowledge cards
Cancel before taking the limit
For ², expanding the numerator gives 2xh+h². For nonzero h the quotient becomes , whose limit is 2x.
Rationalize a radical
For =√x, multiply by the conjugate to cancel h. At the limit yields ; the formula does not give a finite derivative at .
Explore the knowledge in this video
- Derivatives ExplanationAt 0:00
Why this connection?
The derivative is when this finite limit exists. Substitute everywhere in the original expression. For ², expanding the numerator gives 2xh+h². For nonzero h the quotient becomes , whose limit is 2x. For =√x, multiply by the conjugate to cancel h. At the limit yields ; the formula does not give a finite derivative at .
Similar explanations
YouTube2:39Calculus·English
Product rule | Derivative rules | AP Calculus AB | Khan AcademyFoundationalReviewed notes
Another explanation of Derivatives.
Derivatives
YouTube3:53Calculus·English
Power rule | Derivative rules | AP Calculus AB | Khan AcademyFoundationalReviewed notes
Another explanation of Derivatives.
Derivatives
Connect the next step
YouTube4:16Calculus·English
Ex 7: Second Fundamental Theorem of Calculus with Chain RuleIntermediateReviewed notes
Next in Calculus: Derivatives → Chain rule.
Chain rule · View learning sequence
YouTube3:55Next in Calculus: Derivatives → Chain rule.
Chain rule · View learning sequence
YouTube2:26Calculus·English
One-sided limits from graphs: asymptote | Limits and continuity | AP Calculus AB | Khan AcademyFoundationalReviewed notes
Prerequisite in Calculus: Limits → Derivatives.
Limits · View learning sequence
A different approach
Bilibili5:38Visual explanation for Derivatives, compared with the current whiteboard.
Derivatives
YouTube3:58Calculus·English
Derivative intuition module | Taking derivatives | Differential Calculus | Khan AcademyIntroductoryReviewed notes
Visual explanation for Derivatives, compared with the current whiteboard.
Derivatives