What Are the Chain Rule and Product Rule?

When you need to take the derivative of more complicated functions—like a function inside another function, or two functions multiplied together—you use these rules!

The Chain Rule

Used when differentiating a "function within a function," such as \( \sin(x^2) \).

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

The Product Rule

Used when differentiating the product of two functions.

\[ \frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]

Why Are These Rules Important?

They let you tackle more interesting and realistic problems where functions are combined or nested.

When Should You Use Them?

• Chain Rule: Whenever one function is inside another.
• Product Rule: Whenever two functions are multiplied.