Derivatives & Tangent Lines Lab

Explore derivatives visually by watching secant lines converge to tangent lines as h approaches 0. Work through power, product, quotient, and chain rule derivations step by step, and compute slopes on implicit curves using implicit differentiation.

Guided Experiment: Secant to Tangent

How does the secant line change as h approaches 0? Does the secant slope always approach a single value?

Write your hypothesis in the Lab Report panel, then click Next.

Controls

Results

Limit Definition
Function Value
f(1) = 1
Derivative (Slope)
f'(1) = 2
Tangent Line
y = 2x − 1
Secant Line (h = 1)
y = 3x − 2
Secant slope: 3 | Tangent slope: 2 | Difference: 1

Data Table

(0 rows)
#TrialFunctionxf(x)SlopeTangent LineSecant Slope
0 / 500
0 / 500
0 / 500

Reference Guide

Limit Definition of Derivative

The derivative of f at a point a is defined as the limit of the difference quotient.

f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

Geometrically, this is the slope of the tangent line to the graph at x = a, obtained as the limit of secant line slopes.

Power, Product & Quotient Rules

Three fundamental rules for computing derivatives of common function forms.

Power: ddx[xn]=nxn−1\textbf{Power: } \frac{d}{dx}[x^n] = nx^{n-1}
Product: (uv)′=u′v+uv′\textbf{Product: } (uv)' = u'v + uv'
Quotient: (uv)′=u′v−uv′v2\textbf{Quotient: } \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Chain Rule

The chain rule handles the derivative of a composite function f(g(x)).

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function. This applies to any nested composition of functions.

Implicit Differentiation

When y is defined implicitly by F(x,y) = 0, differentiate both sides with respect to x and solve for dy/dx.

dydx=−FxFy=−∂F/∂x∂F/∂y\frac{dy}{dx} = -\frac{F_x}{F_y} = -\frac{\partial F/\partial x}{\partial F/\partial y}

This technique lets you find slopes on curves like circles and ellipses that cannot be written as a single function y = f(x).