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 cc, ddx(c)=0ddx(c)=0
  • Power Rule:
    • For f(x)=xnf(x)=xn where nn is a positive integer: ddx(xn)=nxn1ddx(xn)=nxn1
  • Sum and Difference Rules:
    • ddx(f(x)+g(x))=f(x)+g(x)ddx(f(x)+g(x))=f(x)+g(x)
    • ddx(f(x)g(x))=f(x)g(x)ddx(f(x)g(x))=f(x)g(x)
  • Constant Multiple Rule:
    • For any constant kk, ddx(kf(x))=kf(x)ddx(kf(x))=kf(x)

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 g(x)=3g(x)=3.

Find ddx(x4)ddx(x4)

Find the derivative of f(x)=x7f(x)=x7.

Find the derivative of f(x)=2x36x2+3f(x)=2x36x2+3.

Find the equation of the line tangent to the graph of f(x)=3x211f(x)=3x211 at x=2x=2. Use the point-slope form.

The Advanced Rules

The Main Idea 

  • Product Rule:
    • For j(x)=f(x)g(x)j(x)=f(x)g(x): j(x)=f(x)g(x)+g(x)f(x)j(x)=f(x)g(x)+g(x)f(x)
  • Quotient Rule:
    • For j(x)=f(x)g(x)j(x)=f(x)g(x): j(x)=f(x)g(x)g(x)f(x)[g(x)]2j(x)=f(x)g(x)g(x)f(x)[g(x)]2
  • Extended Power Rule:
    • For kk a negative integer: ddx(xk)=kxk1ddx(xk)=kxk1

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 j(x)=2x5(4x2+x)j(x)=2x5(4x2+x).

Find the derivative of g(x)=1x7g(x)=1x7 using the extended power rule.

Find the derivative of h(x)=(x3+2x)(4x23)h(x)=(x3+2x)(4x23).

Find the derivative of k(x)=x2+3x2x1.

Find the derivative of f(x)=5x32x1.

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 k(x)=f(x)g(x)h(x), express k(x) in terms of f(x),g(x),h(x), and their derivatives.

Find ddx(3f(x)2g(x)).

Find the value(s) of x for which the line tangent to the graph of f(x)=4x23x+2 is parallel to the line y=2x+3.