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.

$f(x)g'(x)$ 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 differentiating a function that is explicitly defined as the product of two other functions.

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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: $g'(x) = \sqrt{x}e^x + e^x(\frac{1}{2\sqrt{x}})$.

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: $h'(x) = (x^2 + 1)(3x^2) + (x^3 - 1)(2x)$.

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