Product Rule

• The Product Rule is applied when a function $y$ is the product of two other functions, typically denoted as $u$ and $v$, both of which are functions of the same variable (usually $x$).

• In Leibniz notation, if $y = uv$, the derivative is expressed as:

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

• In function notation, if $f(x) = g(x)h(x)$, the derivative is:

$f'(x) = g(x)h'(x) + h(x)g'(x)$

• This formula demonstrates that the derivative of a product is NOT simply the product of the derivatives; instead, it is the first function times the derivative of the second, plus the second function times the derivative of the first.

• Step 1: Identification: Clearly identify the two distinct functions $u$ and $v$ that are being multiplied together. For example, in $x^2 \sin(x)$, $u = x^2$ and $v = \sin(x)$.

• Step 2: Individual Differentiation: Differentiate $u$ to find $u'$ (or $\frac{du}{dx}$) and differentiate $v$ to find $v'$ (or $\frac{dv}{dx}$) separately.

• Step 3: Substitution: Plug these four components ($u, v, u', v'$) into the product rule formula $uv' + vu'$.

• Step 4: Simplification: Expand the resulting expression and collect like terms. In many exam contexts, factoring out common terms (like $e^x$ or powers of $x$) is expected for the final answer.

• It is vital to distinguish between a product of functions and a composite function. The Product Rule is for $f(x) \cdot g(x)$, while the Chain Rule is for $f(g(x))$.

• Unlike the Sum Rule, where you simply add the derivatives of individual terms, the Product Rule requires cross-multiplication of the original functions with their respective derivatives.

• Use Brackets: Always place $u, v, u',$ and $v'$ in brackets when substituting into the formula to avoid sign errors, especially when dealing with negative coefficients or trigonometric derivatives.

• Identify the 'Main' Rule: If a function looks complex, determine if the overall structure is a product. Sometimes a Product Rule problem contains a Chain Rule within one of its parts (e.g., $x^2 e^{3x}$).

• Sanity Check: If your result is just the product of two derivatives (e.g., $(x^2 \sin x)' = 2x \cos x$), you have made the most common conceptual error. The result must almost always be a sum of two terms.

• The Naive Product Rule: The most frequent mistake is assuming $(uv)' = u'v'$. This ignores the fundamental logic of how products change and will result in a loss of all marks for the differentiation step.

• Incorrect Identification: Students sometimes apply the product rule to a constant multiplied by a function (e.g., $5x^3$). While technically possible (where $u=5$), it is inefficient; the Constant Multiple Rule is preferred.

• Simplification Errors: Forgetting to apply the distributive law correctly when combining $uv'$ and $vu'$ can lead to incorrect final expressions even if the calculus was performed correctly.