The Knowledge Web

Composite Functions

Imagine a function inside a function: $y = \sin(x^2)$ . You can't just say the derivative is $\cos(2x)$ . You have to account for how fast the inside is changing too.

Rate Multiplier

y = sin(x²)

1. The Trigger (x)

2.0

Inner Rate

g'(x) = 2x

4.00x

Outer Rate

f'(u) = \cos(u)

-0.65x

Total Chain Reaction

4.00-0.65

-2.61

Rate of Change (dy/dx)

The Formula

The rule states that the total rate of change is the product of the individual rates.

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Outer Derivative × Inner Derivative

The "Onion" Strategy

To solve these problems, work from the outside in.

Step 1: Identify Layers

For $y = (3x+1)^5$ :

Outer: $(\dots)^5$
Inner: $3x+1$

Step 2: Differentiate Layers

Outer Rate: $5(\dots)^4$
Inner Rate: $3$

Total:

15(3x+1)^4