Glossary

The is a key tool for differentiating functions that are multiplied together. It breaks down complex expressions into simpler parts, making it easier to find derivatives of tricky functions.

Using this rule, you can tackle derivatives of products involving polynomials, trig functions, and exponentials. It's essential to know when to use it and how to combine it with other rules for more complex problems.

The Product Rule

Product rule for differentiation

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  • States that the derivative of the product of two differentiable functions f(x)f(x) and g(x)g(x) equals:
    • f(x)f(x) times the derivative of g(x)g(x), plus
    • g(x)g(x) times the derivative of f(x)f(x)
  • Formally written as: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]\frac{d}{dx}[f(x)g(x)] = f(x)\frac{d}{dx}[g(x)] + g(x)\frac{d}{dx}[f(x)]
  • Useful for finding derivatives of products of functions (polynomials, , exponentials)
  • To apply the rule:
    1. Identify the two functions being multiplied
    2. Find the derivative of each function separately
    3. Multiply the first function by the derivative of the second
    4. Multiply the second function by the derivative of the first
    5. Add the resulting terms together

Application of product rule

  • Example 1: Derivative of f(x)=x2sin⁡(x)f(x) = x^2 \sin(x)
    • Let u=x2u = x^2 and v=sin⁡(x)v = \sin(x)
    • ddx[u]=2x\frac{d}{dx}[u] = 2x and ddx[v]=cos⁡(x)\frac{d}{dx}[v] = \cos(x)
    • : ddx[f(x)]=x2cos⁡(x)+2xsin⁡(x)\frac{d}{dx}[f(x)] = x^2 \cos(x) + 2x \sin(x)
  • Example 2: Derivative of g(x)=(3x+1)(x3−2x)g(x) = (3x + 1)(x^3 - 2x)
    • Let u=3x+1u = 3x + 1 and v=x3−2xv = x^3 - 2x
    • ddx[u]=3\frac{d}{dx}[u] = 3 and ddx[v]=3x2−2\frac{d}{dx}[v] = 3x^2 - 2
    • Applying the product rule: ddx[g(x)]=(3x+1)(3x2−2)+(x3−2x)(3)\frac{d}{dx}[g(x)] = (3x + 1)(3x^2 - 2) + (x^3 - 2x)(3)
  • Can be applied repeatedly for products of more than two functions

Product rule vs other rules

  • Use the product rule when:
    • The function is a product of two or more differentiable functions
    • The functions being multiplied are not composed of one another
  • Do not use the product rule when:
    • The function is a sum or difference of terms (use sum rule)
    • The function is a quotient of two functions (use )
    • The function is a composition of functions (use )
    • The function is a single term (use basic differentiation rules)

Combining product rule with other rules

  • Break complex expressions into simpler components
    • Identify products, quotients, compositions, sums/differences
    • Apply appropriate rule to each component
    • Combine results using rules for sums, differences, constant multiples
  • Example: Derivative of h(x)=(2x+1)(x2−3)3h(x) = (2x + 1)(x^2 - 3)^3
    • Let u=2x+1u = 2x + 1 and v=(x2−3)3v = (x^2 - 3)^3
    • ddx[u]=2\frac{d}{dx}[u] = 2 (basic differentiation rule)
    • To find ddx[v]\frac{d}{dx}[v], use chain rule:
      • Let w=x2−3w = x^2 - 3, then v=w3v = w^3
      • ddx[w]=2x\frac{d}{dx}[w] = 2x (basic differentiation rule)
      • ddx[v]=3w2ddx[w]=3(x2−3)2(2x)\frac{d}{dx}[v] = 3w^2 \frac{d}{dx}[w] = 3(x^2 - 3)^2(2x)
    • Apply product rule: ddx[h(x)]=(2x+1)(3(x2−3)2(2x))+((x2−3)3)(2)\frac{d}{dx}[h(x)] = (2x + 1)(3(x^2 - 3)^2(2x)) + ((x^2 - 3)^3)(2)

Key Terms to Review (16)

Applying the Product Rule: Applying the product rule is a fundamental technique in differential calculus used to differentiate functions that are products of two or more functions. It states that if you have two functions, say u(x) and v(x), their derivative is given by the formula: $$(u imes v)\' = u\' \times v + u \times v\'$$. This rule is essential for solving problems involving multiplication of functions, enabling students to find rates of change for complex expressions efficiently.
Chain Rule: The chain rule is a fundamental technique in calculus used to differentiate composite functions, allowing us to find the derivative of a function that is made up of other functions. This rule is crucial for understanding how different rates of change are interconnected and enables us to tackle complex differentiation problems involving multiple layers of functions.
Continuity: Continuity in mathematics refers to a property of a function where it does not have any breaks, jumps, or holes over its domain. A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is crucial because it ensures that the behavior of functions can be analyzed smoothly, impacting several important mathematical principles and theorems.
Derivative of a product: The derivative of a product refers to the rule used to differentiate the product of two functions. This is essential in calculus because it allows us to find the rate of change of a product function, which is crucial when analyzing complex relationships between variables. Understanding this concept also leads to deeper insights into how different functions interact and provides a foundation for more advanced topics like higher-order derivatives and applications in physics and engineering.
Derivative of f(x) = x^2 sin(x): The derivative of the function f(x) = x^2 sin(x) represents the rate of change of the function at any given point. To find this derivative, we use the product rule, which allows us to differentiate functions that are products of two simpler functions. In this case, the two functions are g(x) = x^2 and h(x) = sin(x), and applying the product rule helps us understand how the combination of these functions behaves when changes occur.
Derivative of g(x) = (3x + 1)(x^3 - 2x): The derivative of a function represents the rate at which the function value changes with respect to changes in its input. In this context, the product rule is used to find the derivative of the function g(x), which is expressed as the product of two separate functions: (3x + 1) and (x^3 - 2x). Understanding how to apply the product rule allows you to differentiate products of functions effectively, revealing how the combined behavior of those functions influences their overall rate of change.
Differentiation: Differentiation is the process of finding the derivative of a function, which measures how a function changes as its input changes. It provides insights into the rate of change and helps to understand various concepts such as slopes of tangent lines, rates of change in related quantities, and finding maximum or minimum values. This fundamental tool in calculus is essential for analyzing and interpreting mathematical relationships across various contexts.
Exponential Functions: Exponential functions are mathematical functions of the form $$f(x) = a imes b^{x}$$, where $$a$$ is a constant, $$b$$ is a positive real number, and $$x$$ is the exponent. They describe processes that grow or decay at a constant rate proportional to their current value, making them crucial in modeling real-world phenomena such as population growth and radioactive decay.
F'(x): The notation f'(x) represents the derivative of the function f at the point x, indicating the rate at which the function's value changes as x changes. This concept is crucial for understanding how functions behave, particularly in determining slopes of tangent lines, rates of change, and overall function behavior, which are foundational in various applications such as motion analysis and optimization problems.
G'(x): The notation g'(x) represents the derivative of the function g with respect to the variable x. It indicates the rate of change of g at any point x and provides insight into the function's behavior, such as identifying critical points and determining whether the function is increasing or decreasing. This concept becomes particularly important when applying techniques like the product and quotient rules to find derivatives of more complex functions that are products or ratios of simpler functions.
Higher-order derivatives: Higher-order derivatives are the derivatives of a function taken more than once. They provide insights into the behavior of a function by showing how its rate of change itself changes, offering deeper understanding of its curvature and concavity. Higher-order derivatives play a critical role in various mathematical applications, including optimization and curve sketching, and are especially relevant when using rules such as the product rule and when dealing with composite functions.
Linearity of Differentiation: Linearity of differentiation refers to the principle that the derivative of a sum of functions is equal to the sum of their derivatives, and the derivative of a constant multiplied by a function is equal to that constant multiplied by the derivative of the function. This concept highlights how differentiation behaves in a predictable manner with respect to addition and scalar multiplication, allowing for more straightforward calculations when finding derivatives of complex expressions.
Polynomial Functions: Polynomial functions are mathematical expressions that represent relationships involving variables raised to whole number powers, where the coefficients can be real or complex numbers. They are continuous and smooth across their domain, making them crucial in calculus for understanding derivatives, integrals, and behavior of functions.
Product Rule: The product rule is a fundamental principle in calculus that provides a method for finding the derivative of the product of two functions. It states that if you have two functions, say $$u(x)$$ and $$v(x)$$, the derivative of their product can be calculated using the formula: $$ (uv)' = u'v + uv' $$, where $$u'$$ and $$v'$$ are the derivatives of $$u$$ and $$v$$ respectively. This concept is crucial in understanding how derivatives work when dealing with more complex functions that are products of simpler ones.
Quotient Rule: The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. It states that if you have a function that can be expressed as $$f(x) = \frac{g(x)}{h(x)}$$, where both $$g$$ and $$h$$ are differentiable, then the derivative is given by $$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$$. This rule connects to understanding how rates of change behave in division scenarios, as well as its application alongside other rules such as the product rule and logarithmic differentiation.
Trigonometric Functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, commonly used in various fields like physics, engineering, and computer science. These functions include sine, cosine, tangent, cosecant, secant, and cotangent, and they play a vital role in understanding periodic phenomena. Their properties are essential when discussing derivatives, continuity, and the application of rules in calculus.
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