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AP Calculus AB

Differentiation: Rules, Chain Rule, and Implicit Differentiation

Calculus AB

  • ✓By the end of this lesson students will be able to apply the basic differentiation rules (power, product, quotient, sum/difference, constant multiple) to find derivatives of various functions.
  • ✓By the end of this lesson students will be able to correctly apply the Chain Rule to differentiate composite functions.
  • ✓By the end of this lesson students will be able to perform implicit differentiation to find the derivative dy/dx for implicitly defined functions.
  • ✓By the end of this lesson students will be able to use implicit differentiation to find the equation of a tangent line to a curve at a given point.

Key concepts

Basic Differentiation Rules

These fundamental rules allow us to find the derivative of common function types and combinations. They are the building blocks for more complex differentiation.

Power Rule

Used to differentiate functions of the form x^n.

d/dx [x^n] = nx^(n-1)
Constant Multiple Rule

A constant factor can be pulled out of the derivative.

d/dx [cf(x)] = c * d/dx [f(x)]
Sum and Difference Rules

The derivative of a sum or difference of functions is the sum or difference of their derivatives.

d/dx [f(x) +/- g(x)] = d/dx [f(x)] +/- d/dx [g(x)]
Product Rule

Used to differentiate the product of two functions.

d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Quotient Rule

Used to differentiate the quotient of two functions.

d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2
Chain Rule

The Chain Rule is used to differentiate composite functions, which are functions within functions. If y = f(u) and u = g(x), then y is a composite function of x. The rule states that the derivative of the outer function is multiplied by the derivative of the inner function.

d/dx [f(g(x))] = f'(g(x)) * g'(x) or dy/dx = dy/du * du/dx
Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined in terms of one variable (e.g., y = f(x)). Instead, the relationship between x and y is given by an equation where y is implicitly a function of x. When differentiating terms involving y, we must apply the Chain Rule, treating y as an inner function of x, resulting in a dy/dx factor.

Key facts to remember

  • 1The Power Rule is fundamental: d/dx [x^n] = nx^(n-1).
  • 2The Product Rule is d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x).
  • 3The Quotient Rule is d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2.
  • 4The Chain Rule is for composite functions: d/dx [f(g(x))] = f'(g(x)) * g'(x).
  • 5When using implicit differentiation, differentiate every term with respect to x. For terms involving y, remember to multiply by dy/dx due to the Chain Rule.
  • 6After differentiating implicitly, you must algebraically solve for dy/dx.
  • 7To find the equation of a tangent line, you need a point (x1, y1) and the slope m (which is dy/dx evaluated at that point).

Worked examples

Example 1

Find the derivative of y = (4x^3 - 2x)(x^2 + 5x - 1).

IIdentify f(x) = 4x^3 - 2x and g(x) = x^2 + 5x - 1.
IIFind the derivatives of f(x) and g(x): f'(x) = 12x^2 - 2 and g'(x) = 2x + 5.
IIIApply the Product Rule: dy/dx = f'(x)g(x) + f(x)g'(x).
IVSubstitute the functions and their derivatives: dy/dx = (12x^2 - 2)(x^2 + 5x - 1) + (4x^3 - 2x)(2x + 5).
VExpand and simplify: dy/dx = (12x^4 + 60x^3 - 12x^2 - 2x^2 - 10x + 2) + (8x^4 + 20x^3 - 4x^2 - 10x).
VICombine like terms: dy/dx = 20x^4 + 80x^3 - 18x^2 - 20x + 2.

Answer

dy/dx = 20x^4 + 80x^3 - 18x^2 - 20x + 2

Remember to distribute carefully when expanding the terms.

Example 2

Find the derivative of y = cos^4(3x^2 + 7).

IRewrite the function as y = (cos(3x^2 + 7))^4 to clearly see the composite structure.
IIIdentify the outermost function as u^4, where u = cos(3x^2 + 7). The derivative of u^4 is 4u^3 * du/dx.
IIINow differentiate u = cos(3x^2 + 7). Here, the outer function is cos(v) and the inner function is v = 3x^2 + 7. The derivative of cos(v) is -sin(v) * dv/dx.
IVDifferentiate v = 3x^2 + 7: dv/dx = 6x.
VCombine the derivatives using the Chain Rule: dy/dx = 4(cos(3x^2 + 7))^3 * [-sin(3x^2 + 7)] * (6x).
VISimplify the expression: dy/dx = -24x cos^3(3x^2 + 7) sin(3x^2 + 7).

Answer

dy/dx = -24x cos^3(3x^2 + 7) sin(3x^2 + 7)

The Chain Rule can be applied multiple times for nested functions. Work from the 'outside in'.

Example 3

Given the equation x^3 + y^3 = 6xy, find dy/dx and the equation of the tangent line to the curve at the point (3, 3).

IDifferentiate both sides of the equation with respect to x: d/dx [x^3 + y^3] = d/dx [6xy].
IIDifferentiate x^3: 3x^2.
IIIDifferentiate y^3 using the Chain Rule (y is a function of x): 3y^2 * dy/dx.
IVDifferentiate 6xy using the Product Rule: 6(1*y + x*dy/dx) = 6y + 6x dy/dx.
VCombine the differentiated terms: 3x^2 + 3y^2 dy/dx = 6y + 6x dy/dx.
VIRearrange the equation to isolate terms with dy/dx: 3y^2 dy/dx - 6x dy/dx = 6y - 3x^2.
VIIFactor out dy/dx: dy/dx (3y^2 - 6x) = 6y - 3x^2.
VIIISolve for dy/dx: dy/dx = (6y - 3x^2) / (3y^2 - 6x).
9Simplify the expression for dy/dx by dividing numerator and denominator by 3: dy/dx = (2y - x^2) / (y^2 - 2x).
10To find the slope of the tangent line at (3, 3), substitute x=3 and y=3 into dy/dx: m = (2(3) - 3^2) / (3^2 - 2(3)) = (6 - 9) / (9 - 6) = -3 / 3 = -1.
11Use the point-slope form of a line, y - y1 = m(x - x1): y - 3 = -1(x - 3).
12Simplify to slope-intercept form: y - 3 = -x + 3 => y = -x + 6.

Answer

dy/dx = (2y - x^2) / (y^2 - 2x); Tangent line equation: y = -x + 6

Remember to apply the Chain Rule to every term involving y when differentiating implicitly. Also, be careful with algebraic manipulation when solving for dy/dx.

Common mistakes

  • ✗Forgetting to apply the Chain Rule, especially when differentiating terms like (f(x))^n or sin(g(x)).
  • ✗Incorrectly applying the Product or Quotient Rule, particularly mixing up the order or signs in the Quotient Rule numerator.
  • ✗Forgetting to multiply by dy/dx when differentiating terms involving y during implicit differentiation (e.g., d/dx [y^2] is 2y dy/dx, not just 2y).
  • ✗Algebraic errors when isolating dy/dx after implicit differentiation.
  • ✗Not rewriting functions like sin^2(x) as (sin(x))^2 before applying the Chain Rule, leading to confusion about the 'outer' and 'inner' functions.

Exam tips

  • ★Memorize all basic differentiation rules and the Chain Rule, Product Rule, and Quotient Rule formulas. Flashcards can be helpful.
  • ★Show all steps clearly, especially for implicit differentiation and complex Chain Rule problems. This helps in identifying errors and can earn partial credit.
  • ★Practice identifying the 'outer' and 'inner' functions for the Chain Rule. If a function is nested multiple times, apply the Chain Rule repeatedly from the outside in.
  • ★When performing implicit differentiation, gather all terms containing dy/dx on one side of the equation and all other terms on the other side before factoring out dy/dx.

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