Define the Product Rule.

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

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Define the Product Rule.

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

What does $f(x)$ represent in the Product Rule?

One of the functions being multiplied together.

What does $g(x)$ represent in the Product Rule?

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

What does $f'(x)$ represent in the Product Rule?

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

What does $g'(x)$ represent in the Product Rule?

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

Explain why we need a specific rule for products.

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

What is the significance of correctly applying the Product Rule?

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

What is the relationship between the Product Rule and other differentiation rules?

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

Why is understanding the Product Rule important for AP Calculus?

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

What is a common mistake when using the Product Rule?

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

What is the formula for the Product Rule?

$\frac{d}{dx}(f(x)g(x)) = f(x)g'(x) + g(x)f'(x)$

Express the Product Rule using Leibniz notation.

$\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$

Given $y = uv$ , state the formula for finding $\frac{dy}{dx}$ .

$\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$

If $f(x) = u(x)v(x)$ , what is $f'(x)$ using the Product Rule?

$f'(x) = u(x)v'(x) + v(x)u'(x)$

Write the Product Rule, given functions $f$ and $g$ .

$(fg)' = f'g + fg'$

What is the formula to find the derivative of $y = e^x sin(x)$ ?

$y' = e^x \cos(x) + e^x \sin(x)$

What is the derivative of $y = (3x^2-4x)(2x-1)$ using the Product Rule?

$y' = (3x^2-4x)(2) + (2x-1)(6x-4)$

State the product rule formula for $f(x) = sin(x)(3x^2 - 2x + 5)$ .

$f'(x) = \sin(x)(6x-2) + (3x^2-2x+5)\cos(x)$

Write the general form of the Product Rule.

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

What is the formula for $\frac{d}{dx}[u(x)v(x)]$ ?

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

What is the key difference between the Product Rule and the Quotient Rule?

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

Compare the Chain Rule and the Product Rule.

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

Differentiate between using the Power Rule and the Product Rule.

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

When should you use the Product Rule vs. simplifying the expression first?

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

Compare applying the Product Rule directly versus expanding and then differentiating.

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

What is the difference between the Product Rule and the Sum/Difference Rule?

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

Contrast the application of the Product Rule with the Constant Multiple Rule.

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

Compare the situations where you would use the Product Rule versus logarithmic differentiation.

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

Differentiate between using the Product Rule and the Implicit Differentiation.

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

Compare the application of the Product Rule with the Chain Rule when dealing with composite functions involving products.

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