A method for finding the derivative of a function that is the product of two other functions.

One of the functions being multiplied together.

The other function being multiplied by $f(x)$.

The derivative of the function $f(x)$.

The derivative of the function $g(x)$.

The derivative of a product is NOT the product of the derivatives.

Ensuring accurate calculation of the rate of change of a product of functions.

It can be combined with other rules like the chain rule for more complex functions.

It's a fundamental concept frequently tested in both multiple-choice and free-response questions.

Incorrectly calculating the derivative by only multiplying the derivatives of the individual functions.

Product Rule: for functions multiplied together. Quotient Rule: for functions divided by each other.

Chain Rule: for composite functions (function inside a function). Product Rule: for the product of two functions.

Power Rule: for differentiating $x^n$. Product Rule: for differentiating $f(x)g(x)$.

Use Product Rule when functions are distinct and cannot be easily combined. Simplify if possible to avoid it.

Directly applying the Product Rule is useful when functions are complex. Expanding simplifies differentiation for polynomials.

Product Rule: Differentiates the product of two functions. Sum/Difference Rule: Differentiates the sum or difference of functions.

Product Rule: Applies when two functions are multiplied. Constant Multiple Rule: Applies when a function is multiplied by a constant.

Product Rule: For simple products of functions. Logarithmic Differentiation: For complex products or functions raised to variable powers.

Product Rule: For explicit functions. Implicit Differentiation: For implicit functions where y is not explicitly defined in terms of x.

Product Rule: Deals with the product of functions. Chain Rule: Deals with the composition of functions; both may be needed in complex problems.

The sign of $f(x)g'(x) + g(x)f'(x)$ is positive.

The derivative graph shows the slope of the product function, influenced by the slopes and values of the original functions.

A critical point (local max/min) of the product function $f(x)g(x)$.

By comparing the graph of $f(x)g(x)$ with the combined contributions of $f(x)g'(x)$ and $g(x)f'(x)$.

The derivative, $f'(x)g(x) + f(x)g'(x)$, equals zero at that point.

By observing where $f(x)$ and $g(x)$ are increasing or decreasing, and their respective values.

The net change in the function $f(x)g(x)$ over the given interval.

It is likely to be positive, indicating that $f(x)g(x)$ is also increasing.

It influences the rate at which the slope of the product function changes.

The sign of the derivative will depend on the magnitudes and rates of change of $f(x)$ and $g(x)$.