How to find the derivative of $y = x^2e^x$ ?

Identify $f(x) = x^2$ and $g(x) = e^x$ . Find $f'(x) = 2x$ and $g'(x) = e^x$ . Apply the product rule: $y' = x^2e^x + 2xe^x$ .

Steps to differentiate $f(x) = (x^3 + 1)sin(x)$ ?

Let $u = x^3 + 1$ and $v = \sin(x)$ . Find $u' = 3x^2$ and $v' = \cos(x)$ . Use the formula: $f'(x) = (x^3 + 1)\cos(x) + 3x^2\sin(x)$ .

How do you approach finding $y'$ for $y = (3x^2-4x)(2x-1)$ using the Product Rule?

Identify $f(x) = 3x^2 - 4x$ and $g(x) = 2x - 1$ . Find $f'(x) = 6x - 4$ and $g'(x) = 2$ . Apply the Product Rule: $y' = (3x^2-4x)(2) + (2x-1)(6x-4)$ .

What are the steps to find $f'(x)$ if $f(x) = \sin(x)(3x^2 - 2x + 5)$ ?

Identify $u = \sin(x)$ and $v = 3x^2 - 2x + 5$ . Find $u' = \cos(x)$ and $v' = 6x - 2$ . Use the Product Rule: $f'(x) = \sin(x)(6x - 2) + (3x^2 - 2x + 5)\cos(x)$ .

How to find the derivative of $y = e^x\sin(x)$ ?

Let $u = e^x$ and $v = \sin(x)$ . Find $u' = e^x$ and $v' = \cos(x)$ . Apply the Product Rule: $y' = e^x\cos(x) + e^x\sin(x)$ .

What is the process for finding the derivative of $f(x) = x \cdot \ln(x)$ ?

Identify $u(x) = x$ and $v(x) = \ln(x)$ . Find $u'(x) = 1$ and $v'(x) = \frac{1}{x}$ . Apply the Product Rule: $f'(x) = x(\frac{1}{x}) + \ln(x)(1) = 1 + \ln(x)$ .

How do you differentiate $y = x^2 \cos(x)$ using the Product Rule?

Identify $u(x) = x^2$ and $v(x) = \cos(x)$ . Find $u'(x) = 2x$ and $v'(x) = -\sin(x)$ . Apply the Product Rule: $y' = x^2(-\sin(x)) + \cos(x)(2x) = -x^2\sin(x) + 2x\cos(x)$ .

What steps are involved in finding the derivative of $g(x) = \sqrt{x}e^x$ ?

Identify $u(x) = \sqrt{x} = x^{1/2}$ and $v(x) = e^x$ . Find $u'(x) = \frac{1}{2}x^{-1/2}$ and $v'(x) = e^x$ . Apply the Product Rule: $g'(x) = \sqrt{x}e^x + e^x(\frac{1}{2\sqrt{x}})$ .

How do you apply the Product Rule to find the derivative of $h(x) = (x^2 + 1)(x^3 - 1)$ ?

Identify $u(x) = x^2 + 1$ and $v(x) = x^3 - 1$ . Find $u'(x) = 2x$ and $v'(x) = 3x^2$ . Apply the Product Rule: $h'(x) = (x^2 + 1)(3x^2) + (x^3 - 1)(2x)$ .

How to find $f'(x)$ for $f(x) = (2x + 3)\tan(x)$ ?

Let $u = 2x+3$ and $v = \tan(x)$ . Find $u' = 2$ and $v' = \sec^2(x)$ . Use the formula: $f'(x) = (2x+3)\sec^2(x) + 2\tan(x)$ .

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

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

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 $g(x)$ .

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