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If the graph of $f(x)g(x)$ is increasing, what can you infer about $f'(x)$ and $g'(x)$ ?

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

How does the graph of $f(x)$ and $g(x)$ relate to the graph of $(f(x)g(x))'$ ?

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

What does the x-intercept of the derivative of a product, $(f(x)g(x))'$ , represent?

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

How can you visually confirm the Product Rule using graphs?

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

What does a horizontal tangent on the graph of $f(x)g(x)$ imply about its derivative?

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

How can the graphs of $f(x)$ and $g(x)$ help you predict the behavior of $(f(x)g(x))'$ ?

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

What does the area under the curve of $(f(x)g(x))'$ represent?

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

If $f(x)$ and $g(x)$ are both positive and increasing, what does that suggest about the graph of $(f(x)g(x))'$ ?

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

How does the concavity of $f(x)$ and $g(x)$ affect the graph of $(f(x)g(x))'$ ?

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

If $f(x)$ and $g(x)$ have opposite signs, how does that affect the interpretation of $(f(x)g(x))'$ ?

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

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.

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 $\frac{d}{dx}[u(x)v(x)]$ ?

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

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