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Unit 4: Derivatives of Transcendental Functions and the Chain Rule

Trig, exponential, log, inverse, and hyperbolic functions get their own derivative rules, and the chain rule links every composed function back to them.

14 topics · 55 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

3.5 Derivatives of Sine and Cosine

  1. How it worksHow does the special trig limit for sin(x)/x combine with an angle-sum identity to prove sine's derivative?
  2. Worked exampleFind the derivative of f(x) = 3sin(x) - 5cos(x).
  3. Common mistakeWhy isn't the derivative of cos(x) also positive sin(x)?
Your turn1 of 2
Find the derivative of f(x) = 6sin(x) - 6cos(x).

3.5 Derivatives of the Other Four Trigonometric Functions

  1. How it worksHow does writing tan(x) as sin(x)/cos(x) and using the quotient rule produce sec2(x)?
  2. Worked exampleFind the derivative of f(x) = sec(x) + cot(x).
  3. Common mistakeWhy doesn't the derivative of secant look anything like the derivative of cosecant, even though they're reciprocal cousins?
  4. Worked exampleFind the derivative of f(x) = csc(x)·cot(x).
Your turn1 of 2
Find the derivative of f(x) = 4sec(x) + 4cot(x).

3.6 The Chain Rule

  1. How it worksHow does multiplying an outer derivative by an inner derivative capture a rate of a rate?
  2. Going furtherWhat happens to the chain rule's answer when the inner function has a derivative of zero at that point?
  3. Worked exampleFind the derivative of f(x) = square root of (3x + 1) using the chain rule.
  4. Common mistakeWhy isn't the derivative of sin(x2) just cos(x2)?
  5. Worked exampleFind the derivative of f(x) = (3x2 + 1)5 using the chain rule.
Your turn1 of 2
Find the derivative of f(x) = square root of (4x + 4) using the chain rule.

3.6 The Chain Rule with More Than Two Layers

  1. How it worksHow do you chain the chain rule itself to differentiate a function buried inside a function inside a function?
  2. Worked exampleFind the derivative of f(x) = sin(square root of (x2 + 1)) by working from the outside in.
  3. Common mistakeWhy doesn't skipping the innermost derivative in a triple composition still give the right answer?
  4. Worked exampleFind the derivative of f(x) = cos3(5x) by treating it as a triple composition.
  5. Going furtherWhat happens to the layer count in the chain rule when a function is composed with itself repeatedly?
Your turn1 of 2
Find the derivative of f(x) = cos(4x)3 by treating it as a triple composition.

3.8 Implicit Differentiation

  1. How it worksHow does treating y as a hidden function of x and applying the chain rule to every y term unlock its derivative?
  2. Worked exampleFind dy/dx for the curve x2 + y2 = 25 using implicit differentiation.
  3. Common mistakeWhy isn't the derivative of y2 with respect to x just 2y, the way the derivative of x2 is 2x?
  4. Worked exampleFind dy/dx for the curve x3 + y3 = 6xy using implicit differentiation.
Your turn1 of 2
Find dy/dx for the curve 4x2 + 4y2 = 19 using implicit differentiation.

3.8 Finding a Tangent Line to an Implicit Curve

  1. Worked exampleFind the points on the curve x2 + xy + y2 = 7 where the tangent line is horizontal.
  2. Worked exampleFind the equation of the tangent line to x2 + y2 = 25 at the point (3, 4).
  3. Going furtherWhat happens to implicit differentiation at a point on the curve where dy/dx would need to divide by zero?
Your turn1 of 2
For the circle x2 + y2 = 92, use implicit differentiation to find dy/dx as a function of x and y.

3.7 Derivatives of General Inverse Functions

  1. How it worksHow does reflecting a graph across y = x flip its rise-over-run into run-over-rise at matching points?
  2. Common mistakeWhy isn't the derivative of an inverse function just the reciprocal of the original derivative formula, unchanged?
  3. Worked exampleFind the derivative of the inverse of f(x) = x3 + 2x at the point where f(1) = 3, without solving for the inverse.
Your turn1 of 2
Let f(x) = 6x3, for x > 0. Find the derivative of its inverse function, (f−1)'(x).

3.7 Derivatives of Inverse Trigonometric Functions

  1. How it worksHow does implicit differentiation on y = arcsin(x) and a right triangle turn it into 1 over square root of (1 - x2)?
  2. Worked exampleFind the derivative of f(x) = arctan(3x).
  3. Common mistakeWhy doesn't the derivative of arccos(x) come out positive, given that arccos itself is a decreasing function?
  4. Worked exampleFind the derivative of f(x) = arcsin(x2).
Your turn1 of 2
Find the derivative of f(x) = arctan(4x).

3.9 Derivatives of Exponential Functions

  1. How it worksHow does the limit definition of e explain why ex differentiates to itself while 2x does not?
  2. Worked exampleFind the derivative of f(x) = 5e2x - 3x2.
  3. Worked exampleFind the derivative of f(x) = ex^2 using the chain rule.
  4. Common mistakeWhy isn't the derivative of 2x also just 2x?
  5. Ties togetherHow does the chain rule combine with the exponential derivative rule to differentiate e raised to any function of x?
Your turn1 of 2
Find the derivative of f(x) = 6e3x - 6x2.

3.9 Derivatives of Logarithmic Functions

  1. How it worksHow does implicit differentiation on y = ln(x), rewritten as ey = x, produce the derivative 1/x?
  2. Common mistakeWhy isn't the derivative of log base 10 of x also simply 1/x?
  3. Worked exampleFind the derivative of f(x) = ln(3x2 + 1).
Your turn1 of 2
Find the derivative of f(x) = ln(4x2 + 6).

3.9 Logarithmic Differentiation

  1. How it worksHow does logarithmic differentiation let you differentiate a variable raised to a variable power?
  2. Worked exampleFind the derivative of y = (x2 + 1)3·square root of (x - 1)/(x + 2)4 using logarithmic differentiation.
  3. Worked exampleFind the derivative of y = xx using logarithmic differentiation.
  4. Common mistakeWhy doesn't the power rule alone handle a function like xx, even though it has a variable exponent?
Your turn1 of 2
Find the derivative of y = 4·xx using logarithmic differentiation.

6.9 Defining the Hyperbolic Sine and Cosine Functions

  1. How it worksHow do combinations of ex and e−x get defined as sinh(x) and cosh(x) to build the catenary curve?
  2. Worked exampleFind sinh(0) and cosh(0) directly from their definitions in terms of ex.
  3. Common mistakeWhy isn't the graph of cosh(x) simply the graph of cos(x) stretched out?
  4. Going furtherWhat happens to sinh(x) and cosh(x) as x grows very large in the positive direction?
  5. Ties togetherHow does the hanging-chain catenary connect the exponential function to a shape you can actually see?
Your turn1 of 2
Using the definitions sinh(u) = (eu - e−u)/2 and cosh(u) = (eu + e−u)/2 with u = 4x, write cosh(4x) - sinh(4x) in simplified form, as a function of x.

6.9 The Hyperbolic Identity and the Graph of tanh(x)

  1. How it worksHow does substituting the ex definitions of sinh and cosh into cosh2(x) - sinh2(x) collapse it to 1?
  2. Worked exampleFind tanh(0) and describe what value tanh(x) approaches as x grows without bound.
  3. Common mistakeWhy isn't the identity cosh2(x) - sinh2(x) = 1 the same relationship as the circle identity cos2(x) + sin2(x) = 1?
Your turn1 of 2
Using the definition tanh(u) = (eu - e−u)/(eu + e−u) with u = 4x, write tanh(4x) in terms of exponentials, as a function of x.

6.9 Derivatives of the Hyperbolic Functions

  1. How it worksHow does writing sinh(x) and cosh(x) in terms of ex and e−x reveal that each is the other's derivative?
  2. Common mistakeWhy isn't the derivative of cosh(x) equal to negative sinh(x), the way the derivative of cos(x) is negative sin(x)?
  3. Going furtherWhat happens to the graph of tanh(x) and its derivative as x moves toward positive and negative infinity?
  4. Worked exampleFind the derivative of f(x) = cosh(3x) - 2sinh(x).
Your turn1 of 2
Find the derivative of f(x) = cosh(4x) - 4sinh(x).