The Product Rule is a fundamental principle in calculus that describes how to find the derivative of the product of two functions. It states that if you have two functions, u and v, the derivative of their product can be calculated using the formula $$\frac{d}{dx}[uv] = u'v + uv'$$. This rule is essential for differentiating products of functions effectively, allowing you to break down complex derivatives into manageable parts.
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The Product Rule applies specifically when both functions u and v are differentiable at a certain point.
You must find the derivatives of both functions separately (u' and v') before applying them in the Product Rule formula.
The order of multiplication does not affect the result when using the Product Rule; you can switch u and v without changing the outcome.
This rule is particularly useful in calculus when dealing with polynomial, exponential, logarithmic, and trigonometric functions as products.
When applying the Product Rule, it’s important to carefully write down each term to avoid mistakes in calculations.
Review Questions
How do you apply the Product Rule to differentiate a function that is the product of two polynomials?
To differentiate a product of two polynomials using the Product Rule, first identify your two functions u(x) and v(x). Then, calculate their individual derivatives u' and v'. Finally, plug these into the formula $$\frac{d}{dx}[uv] = u'v + uv'$$, which allows you to compute the derivative by combining these results. This method helps simplify the differentiation process for more complex polynomial expressions.
What would happen if you incorrectly applied the Product Rule by neglecting one of the derivatives?
If you neglect one of the derivatives while applying the Product Rule, your final answer will be incorrect. The essence of the Product Rule lies in using both derivatives (u' and v') appropriately. Omitting either would mean you’re not accounting for how each function contributes to the overall rate of change of their product, leading to an inaccurate representation of the derivative.
Evaluate the expression $$\frac{d}{dx}[x^2 \cdot \sin(x)]$$ using the Product Rule and discuss its significance in understanding function behavior.
To evaluate $$\frac{d}{dx}[x^2 \cdot \sin(x)]$$ using the Product Rule, let u = $$x^2$$ and v = $$\sin(x)$$. Thus, u' = $$2x$$ and v' = $$\cos(x)$$. Applying the Product Rule gives us: $$\frac{d}{dx}[x^2 \cdot \sin(x)] = 2x \cdot \sin(x) + x^2 \cdot \cos(x)$$. This result highlights how both components contribute to changes in their product and allows for deeper analysis of rates of change within functions that combine different types of behaviors like polynomial growth and oscillation.
Related terms
Derivative: The derivative measures how a function changes as its input changes, representing the slope of the tangent line to the function at a given point.