← Calculus I

Unit 3: Derivatives — Definition and Basic Rules

The derivative born from a limit, the shortcuts that make it fast to compute, and what it means for a function to fail to have one.

13 topics · 53 questions

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3.1 The Derivative as the Slope of a Tangent Line

  1. How it worksHow does a secant line rotate into the tangent line as its second point slides into the first?
  2. Worked exampleFind the slope of the secant line to y = x2 from x = 1 to x = 1.1, and compare it with the tangent slope at x = 1.
  3. Common mistakeWhy isn't the derivative simply the slope of the curve itself, since a curve has no single slope?
  4. Worked exampleFind the slope of the tangent line to y = x3 at x = 2 by computing the limit of the secant slopes.
  5. Ties togetherHow does the tangent line's slope at a point connect back to the average rate of change over a shrinking interval?
Your turn1 of 2
Find the equation of the tangent line to y = 4x2 at x = 4, written as y(x).

3.1 The Difference Quotient and the Limit Definition of the Derivative

  1. How it worksHow does the difference quotient turn into the derivative as h shrinks to zero?
  2. Common mistakeWhy isn't dy/dx an ordinary fraction you are free to split apart term by term?
  3. Worked exampleFind f'(x) for f(x) = x2 - 3x directly from the limit definition of the derivative.
  4. Going furtherWhat changes in the definition when you write it with x approaching a instead of h approaching 0?
Your turn1 of 2
Find f'(x) for f(x) = 4x2 - 6x, directly from the limit definition of the derivative.

3.1 Computing a Derivative Directly from the Definition

  1. Worked exampleFind the derivative of f(x) = square root of x using the limit definition and a conjugate trick.
  2. Worked exampleFind the derivative of f(x) = 1/x using the limit definition of the derivative.
  3. Going furtherWhat happens to the difference quotient's algebra when the function being differentiated is a fraction?
Your turn1 of 2
Find the derivative of f(x) = 6·square root of x using the limit definition and a conjugate trick.

3.2 The Derivative as a Function

  1. How it worksHow do you sketch f'(x) by reading the steepness of f(x) at each point across its graph?
  2. Common mistakeWhy isn't the graph of f'(x) just a shifted copy of the graph of f(x)?
  3. Worked exampleDraw the graph of f'(x) given the graph of a function that rises, flattens, then falls.
  4. Going furtherWhat happens to the graph of f'(x) at a point where f(x) has a horizontal tangent line?
Your turn1 of 2
Find f'(x) for f(x) = 4x2 + 4x + 8.

3.2 Where a Derivative Fails to Exist

  1. How it worksHow do a corner, a cusp, and a vertical tangent each break the derivative in a different way?
  2. Common mistakeWhy doesn't a function need a jump in its graph to fail to be differentiable somewhere?
  3. Worked exampleShow why f(x) = |x| has no derivative at x = 0 by comparing its left and right difference quotients.
  4. Going furtherWhat happens to the graph of f'(x) itself at a point where f(x) has a vertical tangent line?
Your turn1 of 2
For f(x) = 4*|x|, find (the left-hand difference quotient at x=0) minus (the right-hand difference quotient at x=0).

3.2 Differentiability Implies Continuity, Not the Reverse

  1. How it worksProve that a function differentiable at a point must also be continuous there.
  2. Common mistakeWhy isn't continuous means differentiable a valid rule, even though differentiable does force continuous?
  3. Worked exampleShow that f(x) = |x| is continuous at x = 0 but not differentiable there.
Your turn1 of 2
Let f(x) = x2 for x < 4, and f(x) = 7(x - 4) + 42 for x ≥ 4 (chosen so f is continuous at x = 4). Find (the left-hand derivative of f at x = 4) minus (the right-hand derivative of f at x = 4).

3.4 Continuity and Differentiability of a Piecewise Function Together

  1. How it worksHow do you check both continuity and matching one-sided derivatives at the same seam of a piecewise function?
  2. Worked exampleFind the value of b that makes f(x) = x2 for x less than 2 and f(x) = bx - 1 for x at least 2 continuous at x = 2.
  3. Worked exampleFind the values of a and b that make f(x) = x2 for x less than 1 and f(x) = ax + b for x at least 1 both continuous and differentiable at x = 1.
  4. Ties togetherHow does checking a piecewise function's seam pull together everything continuity and differentiability separately require?
Your turn1 of 2
Find the value of b that makes f(x) = x2 for x < 5, and f(x) = 6x + b for x ≥ 5, continuous at x = 5.

3.3 The Power Rule

  1. How it worksHow does expanding (x + h)n with the binomial theorem reveal the power rule hiding inside the limit definition?
  2. Common mistakeWhy doesn't the power rule apply directly to something like 2x, even though it also has an exponent?
  3. Worked exampleFind the derivative of g(x) = 1/x2 by rewriting it as a negative power and applying the power rule.
  4. Worked exampleFind the derivative of f(x) = x5 - 4x3 + 7 using the power rule.
  5. Going furtherWhat happens to the power rule when the exponent itself is an irrational number like π?
Your turn1 of 2
Find the derivative of f(x) = 6/x2 by rewriting it as a negative power and applying the power rule.

3.3 The Constant Multiple, Sum, and Difference Rules

  1. How it worksProve the sum rule for derivatives directly from the limit definition of the derivative.
  2. Common mistakeWhy isn't the derivative of a constant function ever anything other than zero?
  3. Worked exampleFind the derivative of f(x) = 7x5 + x3 - 2x + 1/x.
  4. Worked exampleFind the derivative of f(x) = 3x4 - 2x2 + 5x - 9.
Your turn1 of 2
Find the derivative of f(x) = 4x4 - 4x2 + 8x - 3.

3.3 The Product Rule

  1. How it worksHow does splitting a growing rectangle's area into three pieces reveal where the product rule comes from?
  2. Worked exampleFind the derivative of h(x) = (x2 + 1)(x3 - 2x) using the product rule.
  3. Common mistakeWhy doesn't (fg)' equal f'g' the way you might guess by analogy with the sum rule?
  4. Going furtherWhat happens to the product rule when it is applied to three functions multiplied together instead of two?
  5. Worked exampleFind the derivative of h(x) = x3·sin(x) using the product rule.
Your turn1 of 2
Find the derivative of h(x) = (4x2 + 4)·(x3 - 5x) using the product rule.

3.3 The Quotient Rule

  1. How it worksHow do you derive the quotient rule by rewriting f/g as f times g to the power of negative one?
  2. Going furtherWhat happens to the quotient rule when the denominator is just a constant?
  3. Worked exampleFind the derivative of h(x) = (2x - 1)/(x2 + 3) using the quotient rule.
  4. Worked exampleFind the derivative of h(x) = (x2 + 1)/(x - 3) using the quotient rule.
  5. Common mistakeWhy isn't the order of the terms in the quotient rule's numerator something you can swap freely?
Your turn1 of 2
Find the derivative of h(x) = (4x - 4)/(x2 + 6) using the quotient rule.

3.4 Derivatives as Rates of Change in Motion

  1. How it worksHow do position, velocity, and acceleration stack up as a function and its first and second derivatives?
  2. Common mistakeWhy doesn't negative velocity mean an object is slowing down?
  3. Going furtherWhat happens to a position graph at the exact moment an object's velocity passes through zero?
  4. Worked exampleFind the velocity and acceleration at t = 2 for a position function s(t) = t3 - 6t2 + 9t.
Your turn1 of 2
For s(t) = t3 - 6t2 + 6t, find the velocity v(t).

3.2 Higher-Order Derivatives and Jerk

  1. How it worksHow does differentiating repeatedly move a physical quantity from position to velocity to acceleration to jerk?
  2. Worked exampleFind f''(x) and f'''(x) for f(x) = x4 - 5x2 + 2.
  3. Common mistakeWhy isn't the second derivative just a fancier way of writing the first derivative?
Your turn1 of 2
Find f''(x) for f(x) = 4x4 - 6x2 + 8.