Visual Derivative Explanations

See calculus concepts come to life with interactive graphs, animations, and 3D visualizations

📈 Interactive Graphs 🎬 Animated Explanations 🎨 3D Visualizations 🖱️ Click & Drag

🎨 Interactive Visualization Dashboard

Function Selector

Visualization Options

Animation Controls

Speed: Medium

📊 What is a Derivative? (Visual)

Slope of Tangent Line

The derivative at a point is the slope of the tangent line to the curve at that point. As the secant line approaches the tangent line, the slope approaches the derivative.

Visual Process:

  1. Start with two points on the curve
  2. Draw a secant line between them
  3. Move points closer together
  4. Secant becomes tangent line
  5. Slope = Derivative

Interactive Example

Click "Show Tangent Animation" to start
Point A Point B

⛓️ Chain Rule Visualization

Function Composition

The chain rule is easier to understand visually. When you have f(g(x)), you're applying one function to the result of another.

x
Input
g
Inner Function
f
Outer Function
f(g(x))
Output

Example: sin(x²)

Select a step to visualize

Chain Rule: f'(x) = f'(g(x)) · g'(x)

For sin(x²): cos(x²) · 2x

✖️➗ Product & Quotient Rules Visualized

Product Rule: Area Interpretation

Think of u(x) and v(x) as sides of a rectangle. The product u·v is the area. When both sides change, the change in area has two parts.

u
Δu
v
Δv
(u·v)' = u'·v + u·v'

Quotient Rule: Rate of Change

The quotient rule shows how the ratio u/v changes when both numerator and denominator change.

u(x)
/
v(x)
Numerator Change
u'·v
Denominator Change
u·v'
(u/v)' = (u'·v - u·v')/v²

🎨 3D & Multivariable Visualizations

Partial Derivatives

For functions of two variables, partial derivatives show how the function changes when you move in just the x-direction or y-direction.

Surface: f(x,y)
∂f/∂x
∂f/∂y

Gradient Vector Field

The gradient ∇f points in the direction of steepest ascent. The length shows how steep the slope is.

∇f(x,y) = (∂f/∂x, ∂f/∂y)

🎬 Animation Gallery

📈

Secant to Tangent

Watch secant lines become tangent lines

📐

Derivative as Slope

Visualize derivatives as changing slopes

⛓️

Chain Rule Flow

See function composition visually

✖️

Product Rule Area

Area interpretation of product rule

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