Chain Rule & Transcendental Derivatives

Compute derivatives of composite functions. The Chain Rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$, reveals how inner and outer rates of change multiply to dictate the composite slope.

Interactive Sandbox

Derivative Multiplier Grapher

Select a composite function preset. Adjust evaluation slider $x_0$. Observe how the tangent slope of the inner function $g'(x_0)$ (cyan) and the outer function $f'(u_0)$ (rose) multiply to equal the composite slope (indigo).

Inner function: u = g(x) = x²

Outer function: y = f(u) = sin(u)

Composite function: y = f(g(x)) = sin(x²)

Composite Slope Formula

dy/dx = f'(g(x)) • g'(x)

Evaluation Point (x₀) 1.00

Change x₀ to verify that composite slope (Indigo) is always equal to Inner Slope × Outer Slope.

Algebraic Solver

Step-by-Step Solver

Trace the differential substitutions and rates multiplication details for the active parameters.

Chain Rule Expansion Steps

Rates Summary

Inner Rate (du/dx)

g'(1.00) = 2.00

Outer Rate (dy/du)

f'(1.00) = 0.54

Composite Rate (dy/dx)

y' = 1.08

Evaluation

Conceptual Quiz

Test your understanding of derivative chain compositions and transcendental functions.

Question 1 of 5 Score: 0/0

If y = f(g(x)), what is the correct formulation of the Chain Rule?

Correct Answer!

Explanation text...

Calculus Derivatives Rules

sin'(x) = cos(x) | cos'(x) = -sin(x)
(e^x)' = e^x | (ln x)' = 1/x
Chain Rule: dy/dx = (dy/du) * (du/dx)
Power rule: (x^n)' = n x^(n-1)