Calculus: Derivatives
From continuity to inverse hyperbolic functions. Thirty-three short, interactive courses that build up the derivative one idea at a time.
1Continuity
2The derivative
Lesson 3
The Definition of the Derivative
Slide a second point toward the first and watch the secant line turn into the tangent.
Lesson 4
Tangent Lines and Local Linearity
Zoom in on a smooth curve until you can no longer tell it apart from its tangent line.
Lesson 5
Reading Velocity from a Position Graph
The slope of a position graph is velocity: moving forward, moving back, or standing still.
Lesson 6
Continuous but Not Differentiable
Corners, cusps, and vertical tangents: places where a curve stays connected but has no single slope.
Lesson 7
Higher-Order Derivatives
The second derivative tells which way a curve bends. In motion, it is acceleration.
Lesson 8
Derivatives from the Definition and the Power Rule
Expand, cancel the vanishing increment, and spot the pattern behind the power rule.
3Differentiation rules
Lesson 9
The Product Rule
Grow a rectangle and watch its new area come from two thin strips, not from multiplying the rates.
Lesson 10
Where the Quotient Rule’s Minus Sign Comes From
Derive the quotient rule from the product rule and see exactly where the minus sign comes from.
Lesson 11
The Chain Rule
Chain two rates of change together and watch the multipliers stack up layer by layer.
Lesson 12
Combining the Rules
Find the last operation first, then peel the expression apart one rule at a time.
Lesson 13
Differentiating Piecewise Functions
At the seam, check continuity first, then compare the slopes from the left and the right.
4Trigonometric functions
Lesson 14
The Limit of sin θ / θ
Squeeze sin θ / θ between two triangles and a sector to show it approaches 1, but only in radians.
Lesson 15
Rewriting Trig Limits
Reshape unfamiliar trig limits into the basic ones, and see where the cosine limit comes from.
Lesson 16
Derivatives of Sine and Cosine
Record the slope of sine at every point, and the curve you trace is cosine.
Lesson 17
Derivatives of tan, cot, sec, csc
Derive the other four with the quotient rule, and notice which ones pick up a minus sign.
Lesson 18
Simple Harmonic Motion
Differentiate position twice and find an acceleration that always pulls back toward equilibrium.
5Implicit differentiation & related rates
Lesson 19
Implicit Differentiation
Treat y as a function of x, differentiate both sides, and find tangent slopes on a circle.
Lesson 20
Second Implicit Derivative
Differentiate again, substitute the first derivative back, and simplify with the original equation.
Lesson 21
Related Rates: Plug In Last
Relate, differentiate, then plug in. Plug in too early and changing quantities freeze into constants.
Lesson 22
The Sliding Ladder
Slide the base of a ladder out at a steady speed and watch the top fall faster and faster.
6Exponentials & logarithms
Lesson 23
The Number e
Compound interest more and more often and the balance settles toward a limit: e ≈ 2.718.
Lesson 24
The Exponential That Is Its Own Derivative
Every exponential is proportional to its own derivative. Find the base where the constant is exactly 1.
Lesson 25
Derivative of the Natural Log
Rewrite ln x as an exponential, differentiate implicitly, and arrive at 1/x.
Lesson 26
Exponentials and Logs in Any Base
Convert any base to e, and a constant factor, the natural log of the base, appears.
Lesson 27
Logarithmic Differentiation
When the variable sits in both the base and the exponent, take logs before you differentiate.
Lesson 28
Exponential Growth and Decay
When the rate is proportional to the amount, change is exponential and the half-life never changes.
7Inverse & hyperbolic functions
Lesson 29
Hyperbolic Functions
Build sinh and cosh from two exponentials and see why their points land on a hyperbola.
Lesson 30
Derivatives of Inverse Functions
Reflect a graph across the diagonal, and slopes at matching points become reciprocals.
Lesson 31
Defining Inverse Trig Functions
Restrict sine to a one-to-one stretch before inverting it, and see how each range is chosen.
Lesson 32
Derivatives of Inverse Trig Functions
Differentiate implicitly, then use a right triangle to write the answer in terms of x.
Lesson 33
Inverse Hyperbolic Functions
Write inverse hyperbolic functions as logarithms, and their derivatives become plain algebra.