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A. Rate of Change · Derivatives of Exponential and Sinusoidal Functions

Derivatives of Exponential and Sinusoidal Functions: Unit Review

A2.4 A2.5 A2.6 A2.7 A2.8 A3.5About 40 minutes
Learning goals
  • I can differentiate exponential and sinusoidal functions and explain where the rules come from.
  • I can combine these rules with the product, quotient and chain rules.
  • I can solve rate-of-change problems with exponential and sinusoidal models.

Words to know

e
The base, about 2.71828, for which d/dx eˣ = eˣ.
ln x
The natural logarithm, the inverse of eˣ.
derivative cycle
sin x → cos x → −sin x → −cos x → sin x.

Let's review!

In this unit you discovered that an exponential function's derivative is proportional to the function, found the special base e whose function is its own derivative, met the natural logarithm as the inverse of eˣ, and proved d/dx aˣ = aˣ ln a. You then found by investigation that d/dx sin x = cos x and d/dx cos x = −sin x, and combined all of these with the product, quotient and chain rules to solve problems about populations, medicine, cooling, Ferris wheels, tides and daylight.

The next unit, Curve Sketching: Connecting f, f′ and f″, uses these derivatives (and second derivatives) to sketch graphs. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Keep your calculator in radian mode for every trig question. Work one or two sections at a time. Before differentiating, name the structure (product, quotient, or one function inside another). Give exact answers (like 8 ln 2 or e²) when you can, and round only at the end. For each question you miss, write down why (forgot the chain-rule factor? sign error with cosine? confused ln a with log a?) and go back to that lesson. Finish with the 10-question quiz.

functionderivativeeˣeˣaˣaˣ ln ae^(g(x))e^(g(x)) g′(x)sin xcos xcos x−sin xtan xsec² xsin(g(x))cos(g(x)) g′(x)
The core derivative rules of this unit in one table. Every other derivative in the unit comes from these plus the product, quotient and chain rules.
eˣ2ˣsin xe^(−x)solid: f(x)dashed: f′(x)For eˣ the two curves are the same curve.
Each function (solid) with its derivative (dashed). The derivative of 2ˣ is a vertical compression of 2ˣ, the derivative of sin x is cos x, the derivative of e^(−x) is its reflection in the x-axis, and eˣ is its own derivative.

The Derivative of an Exponential Function Is Proportional to the Function

1

For f(x) = 2ˣ, f′(x) ≈ 0.693 × 2ˣ. Find f′(5).

2

For f(x) = aˣ, why is f′(0) called k?

3

Is the graph of the derivative of 1.5ˣ above or below the graph of 1.5ˣ?

Discovering e: The Function That Is Its Own Derivative

1

Differentiate y = 7eˣ − x².

2

Find the equation of the tangent to y = eˣ at x = 1.

3

What does (1 + 1/n)ⁿ approach as n grows?

The Natural Logarithm: ln x as the Inverse of eˣ

1

Evaluate ln(e⁻⁴) and e^(ln 9).

2

Solve 5e^(2x) = 40.

3

State the domain, range and asymptote of y = ln x.

The Derivative of aˣ: Where ln a Comes From

1

Differentiate y = 3ˣ and find the exact slope at x = 2.

2

Is y = 0.6ˣ increasing or decreasing? Use its derivative.

3

Explain in one line why d/dx eˣ = eˣ is a special case of d/dx aˣ = aˣ ln a.

The Chain Rule with Exponential Functions

1

Differentiate y = e^(−4x).

2

Differentiate y = e^(x² + 3x).

3

Differentiate y = 10^(2x).

4

For P(t) = 300e^(0.04t), find P′(t) ÷ P(t).

Products and Quotients with Exponential Functions

1

Differentiate y = x² e^(−x) and factor.

2

Find the horizontal tangent of y = x e^(−2x).

3

Differentiate y = eˣ/(x + 1).

Investigating the Derivative of sin x

1

What is d/dx sin x, and what angle unit does it need?

2

Find the tangent to y = sin x at x = π.

3

Evaluate the limit of (sin h)/h as h → 0.

The Derivative of cos x and the Derivative Cycle

1

Differentiate y = 3 cos x − 2 sin x.

2

Find the 22nd derivative of cos x.

3

s(t) = 6 cos t cm. When is the speed greatest, and what is it?

The Chain Rule with Sinusoidal Functions

1

Differentiate y = 2 sin(5x − 1) + 3.

2

Differentiate y = cos²x.

3

Differentiate y = sin(x³).

4

What is the greatest rate of change of y = 0.5 cos(8x)?

The Derivative of tan x and Other Trigonometric Combinations

1

What is d/dx tan x?

2

Differentiate y = x² sin x.

3

Differentiate y = eˣ sin x and find all x in [0, 2π] with a horizontal tangent.

Solving Problems with Exponential Rates of Change

1

A(t) = 120e^(−0.05t) mg after t hours. Find A′(10) and interpret it.

2

T(t) = 20 + 70e^(−0.1t). When is it cooling at 3.5°C per minute?

3

Why is an exponential model a good fit for a quantity whose growth rate is proportional to its size?

Solving Problems with Sinusoidal Rates of Change

1

h(t) = 20 − 18 cos(πt/2) m, t in minutes. Find the greatest vertical speed.

2

For the tide h(t) = 7 cos(2πt/12.4) + 8, when is the water level not changing?

3

A buoy's height is y = 0.8 sin(1.5t) m. Find y′(0).

Putting It Together: Mixed Functions, Tangent Lines and Checking Derivatives

1

Differentiate y = e^(sin 2x).

2

Find the tangent to y = x + e^(−x) at x = 0.

3

Find and fix the error: d/dx e^(x³) = 3x² e^(3x²).

Just for fun

Unit summary in one sentence: eˣ never changes when you differentiate it, sin x needs four tries to get back home, and everything else just needs the chain rule.

Check your understanding

Quiz

1.

d/dx eˣ =

2.

d/dx 5ˣ =

3.

Solve e^(3x) = 12.

4.

d/dx e^(x² + 1) =

5.

d/dx (x eˣ) =

6.

d/dx cos x =

7.

d/dx sin(3x) =

8.

What is the 6th derivative of sin x?

9.

The greatest rate of change of h(t) = 15 sin(πt/30) + 20 is:

10.

P(t) = 2000e^(0.03t). P′(10) is closest to:

For parents and teachers

Use the review to find what to revisit:

  • Confusing the derivative of aˣ with the power rule, or unsure why f′ = k f → Lesson 1 (The Derivative of an Exponential Function Is Proportional to the Function).
  • Not sure why e is special, or tangent lines to eˣ → Lesson 2 (Discovering e).
  • Errors solving equations with e and ln, or log-law mistakes → Lesson 3 (The Natural Logarithm).
  • Forgetting ln a, or wrong sign for decay bases → Lesson 4 (The Derivative of aˣ).
  • Missing the chain-rule factor in e^(g(x)) → Lesson 5 (The Chain Rule with Exponential Functions).
  • Product or quotient rule slips, or trying to solve eˣ = 0 → Lesson 6 (Products and Quotients with Exponential Functions).
  • Degree-mode answers, or unsure why d/dx sin x = cos x → Lesson 7 (Investigating the Derivative of sin x).
  • Sign errors with cosine, or higher derivatives → Lesson 8 (The Derivative of cos x and the Derivative Cycle).
  • Forgetting k in sin(kx), or confusing sin² x and sin(x²) → Lesson 9 (The Chain Rule with Sinusoidal Functions).
  • tan x and trig products or quotients → Lesson 10 (The Derivative of tan x and Other Trigonometric Combinations).
  • Interpreting growth, decay and cooling rates, or solving for t → Lesson 11 (Solving Problems with Exponential Rates of Change).
  • Ferris wheel, tide and daylight problems → Lesson 12 (Solving Problems with Sinusoidal Rates of Change).
  • Choosing rules for mixed functions, or checking answers numerically → Lesson 13 (Putting It Together).

Looking ahead: the next unit, Curve Sketching: Connecting f, f′ and f″, uses first and second derivatives (including these new ones) to find increasing and decreasing intervals, maximums, minimums and concavity. Applications of Derivatives then uses them for motion and optimization problems.