Math Problem Statement

What are the useful rules about derivatives including linearity and the product rule?

Solution

The image contains some useful rules about derivatives, particularly focusing on linearity and the product rule. Let me summarize the content for you:

Linearity of Derivatives:

  1. Sum Rule: If f(x)f(x) and g(x)g(x) are differentiable at the point aa, then: (f+g)(a)=f(a)+g(a)(f + g)'(a) = f'(a) + g'(a)

  2. Scalar Multiplication Rule: If f(x)f(x) is differentiable at the point aa and cRc \in \mathbb{R}, then: (cf)(a)=cf(a)(cf)'(a) = c f'(a)

  3. Difference Rule: If f(x)f(x) and g(x)g(x) are differentiable at the point aa, then: (fg)(a)=f(a)g(a)(f - g)'(a) = f'(a) - g'(a)

The Product Rule:

If f(x)f(x) and g(x)g(x) are differentiable at aa, then: (fg)(a)=f(a)g(a)+f(a)g(a)(fg)'(a) = f(a)g'(a) + f'(a)g(a) The proof is outlined using the definition of a derivative, starting with: limh0f(a+h)g(a+h)f(a)g(a)h\lim_{h \to 0} \frac{f(a+h)g(a+h) - f(a)g(a)}{h}


Would you like further explanations or a worked example of one of these rules? Here are some follow-up questions:

  1. What is the practical application of the product rule in solving calculus problems?
  2. How can the sum or difference rules simplify differentiating polynomial functions?
  3. Can you explain how these rules extend to higher derivatives (e.g., second derivatives)?
  4. Could you show an example where both the product and sum rules are used together?
  5. How does the proof of the product rule reinforce the importance of the limit definition of a derivative?

Tip: Mastering the product rule is essential for understanding more complex rules, such as the chain rule and quotient rule!

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Linearity of Derivatives
Product Rule

Formulas

(f + g)'(a) = f'(a) + g'(a)
(cf)'(a) = c f'(a)
(f - g)'(a) = f'(a) - g'(a)
(fg)'(a) = f(a)g'(a) + f'(a)g(a)

Theorems

Sum Rule
Scalar Multiplication Rule
Difference Rule
Product Rule

Suitable Grade Level

Grades 11-12, College level