Differentiation Rules: Fresh Take

  • Use basic rules to differentiate functions that involve adding, subtracting, scaling, and raising to powers
  • Apply specific rules to find derivatives of functions multiplied or divided by each other
  • Use a combination of rules to calculate derivatives for polynomial and rational functions

The Basic Rules

The Main Idea 

  • Constant Rule:
    • For any constant [latex]c[/latex], [latex]\frac{d}{dx}(c) = 0[/latex]
  • Power Rule:
    • For [latex]f(x) = x^n[/latex] where [latex]n[/latex] is a positive integer: [latex]\frac{d}{dx}(x^n) = nx^{n-1}[/latex]
  • Sum and Difference Rules:
    • [latex]\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)[/latex]
    • [latex]\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)[/latex]
  • Constant Multiple Rule:
    • For any constant [latex]k[/latex], [latex]\frac{d}{dx}(kf(x)) = kf'(x)[/latex]

Key Concepts

  • These rules form the foundation for differentiating more complex functions.
  • The Power Rule applies to positive integer exponents and will be extended to other exponents later.
  • These rules allow us to differentiate polynomials and many other functions without using the limit definition every time.

Find the derivative of [latex]g(x)=-3[/latex].

Find [latex]\frac{d}{dx}(x^4)[/latex]

Find the derivative of [latex]f(x)=x^7[/latex].

Find the derivative of [latex]f(x)=2x^3-6x^2+3[/latex].

Find the equation of the line tangent to the graph of [latex]f(x)=3x^2-11[/latex] at [latex]x=2[/latex]. Use the point-slope form.

The Advanced Rules

The Main Idea 

  • Product Rule:
    • For [latex]j(x) = f(x)g(x)[/latex]: [latex]j'(x) = f'(x)g(x) + g'(x)f(x)[/latex]
  • Quotient Rule:
    • For [latex]j(x) = \frac{f(x)}{g(x)}[/latex]: [latex]j'(x) = \frac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2}[/latex]
  • Extended Power Rule:
    • For [latex]k[/latex] a negative integer: [latex]\frac{d}{dx}(x^k) = kx^{k-1}[/latex]

Key Concepts

  • The product rule is not simply the product of the derivatives.
  • The quotient rule involves a specific arrangement of terms in the numerator.
  • The extended power rule allows differentiation of negative integer powers.
  • These rules expand our ability to differentiate more complex functions.

Use the product rule to obtain the derivative of [latex]j(x)=2x^5(4x^2+x)[/latex].

Find the derivative of [latex]g(x)=\dfrac{1}{x^7}[/latex] using the extended power rule.

Find the derivative of [latex]h(x) = (x^3 + 2x)(4x^2 - 3)[/latex].

Find the derivative of [latex]k(x) = \frac{x^2 + 3x}{2x - 1}[/latex].

Find the derivative of [latex]f(x) = 5x^{-3} - 2x^{-1}[/latex].

Combining Differentiation Rules

The Main Idea 

  • Combining Multiple Rules:
    • Most real-world problems require applying several differentiation rules in sequence
    • Apply rules in reverse order of function evaluation
  • Problem-Solving Strategy:
    • Identify the structure of the function
    • Determine which rules apply and in what order
    • Apply the rules systematically
    • Simplify the result
  • The order of applying differentiation rules matters
  • Complex functions often require a combination of product, quotient, and basic rules

For [latex]k(x)=f(x)g(x)h(x)[/latex], express [latex]k^{\prime}(x)[/latex] in terms of [latex]f(x), \, g(x), \, h(x)[/latex], and their derivatives.

Find [latex]\frac{d}{dx}(3f(x)-2g(x))[/latex].

Find the value(s) of [latex]x[/latex] for which the line tangent to the graph of [latex]f(x)=4x^2-3x+2[/latex] is parallel to the line [latex]y=2x+3[/latex].