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What is the formula for the Product Rule?

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

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

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

Explain why you cannot simply multiply the derivatives of two functions to find the derivative of their product.

The derivative represents the instantaneous rate of change, and the rate of change of a product is influenced by both functions' rates and their interaction.

Describe the role of each term in the Product Rule formula.

$f(x)g'(x)$ represents the contribution of $f(x)$ to the rate of change, scaled by the rate of change of $g(x)$ , and vice versa for $g(x)f'(x)$ .

When is the Product Rule essential?

When differentiating a function that is explicitly defined as the product of two other functions.

What does the Product Rule tell us about the derivative of a product?

The derivative of a product involves both the original functions and their derivatives, combined in a specific way.

Explain the concept behind 'first d second plus second d first'.

It's a mnemonic to remember the product rule: (First function) * (Derivative of second) + (Second function) * (Derivative of first).

Why is the Product Rule considered a high-value topic in calculus?

It is frequently used and tested, often in combination with other differentiation rules.

What does the Product Rule help us find?

The instantaneous rate of change of a product of two functions.

How does the Product Rule relate to the concept of rates of change?

It accounts for how the rates of change of two functions combine when they are multiplied together.

What is the significance of the plus sign in the Product Rule?

It indicates that we are summing the contributions of each function's derivative to the overall derivative of the product.

Explain why the order of terms in the Product Rule doesn't matter.

Because addition is commutative, $f(x)g'(x) + g(x)f'(x)$ is the same as $g(x)f'(x) + f(x)g'(x)$ .

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 $g(x)$ .