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

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

A3.5 A2.4 A2.8About 70 minutes

Learning goals

I can choose and combine the power, product, quotient and chain rules for any combination of polynomial, exponential and sinusoidal functions.
I can find equations of tangent lines and the locations of horizontal tangents for mixed functions.
I can check a derivative numerically with a centred difference quotient on a calculator.
I can find and fix common errors in differentiation.

Words to know

structure of a function
The order in which operations are applied. The operation done last decides which rule you use first.

centred difference quotient
[f(a + h) − f(a − h)] ÷ (2h), a very accurate numerical estimate of f′(a) for small h.

tangent line
y − f(a) = f′(a)(x − a), the line touching the curve at x = a.

normal line
The line through a point on a curve perpendicular to the tangent. Its slope is −1/f′(a) when f′(a) ≠ 0.

damped oscillation
A wave whose amplitude decreases, such as e^(−0.5x) sin x.

Your full toolkit

You now have every rule this course asks for. Here they are in one place:

Power: d/dx xⁿ = nxⁿ⁻¹ (any rational n)
Exponential: d/dx eˣ = eˣ and d/dx aˣ = aˣ ln a
Sinusoidal: d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec² x
Product, quotient and chain rules, plus the constant, constant multiple, sum and difference rules from Derivatives and Differentiation Rules.

The hard part is no longer the rules. It is seeing the structure of a function so you apply them in the right order.

To choose a rule, ask what operation is done last when you evaluate the function. That decides the first rule; then repeat the question for each piece.

Worked example

Example 1: Choosing the rules

Differentiate (a) f(x) = x² e^(3x) (b) g(x) = sin(e^(2x)) (c) h(x) = (1 + cos x)/eˣ

(a) Last step: multiply. Product rule, with a chain rule inside: f′(x) = 2x e^(3x) + x² × 3e^(3x) = x e^(3x)(2 + 3x).
(b) Last step: sine of something. Chain rule twice: g′(x) = cos(e^(2x)) × e^(2x) × 2 = 2e^(2x) cos(e^(2x)).
(c) Last step: divide. Rewrite as a product, h(x) = (1 + cos x)e^(−x): h′(x) = (−sin x)e^(−x) + (1 + cos x)(−e^(−x)) = −e^(−x)(1 + sin x + cos x).

(a) x e^(3x)(2 + 3x) (b) 2e^(2x) cos(e^(2x)) (c) −e^(−x)(1 + sin x + cos x)

Worked example

Example 2: A tangent line

Find the equation of the tangent to y = eˣ sin x at x = π/2.

Point: y(π/2) = e^(π/2) sin(π/2) = e^(π/2) ≈ 4.810.
y′ = eˣ sin x + eˣ cos x = eˣ(sin x + cos x).
Slope: e^(π/2)(1 + 0) = e^(π/2).
y − e^(π/2) = e^(π/2)(x − π/2), so y = e^(π/2)(x + 1 − π/2) ≈ 4.810x − 2.746.

y = e^(π/2)(x + 1 − π/2), about y = 4.810x − 2.746.

Worked example

Example 3: Horizontal tangents of a damped wave

Find all points in [0, 2π] where y = e^(−0.5x) sin x has a horizontal tangent.

Product and chain rules: y′ = −0.5e^(−0.5x) sin x + e^(−0.5x) cos x = e^(−0.5x)(cos x − 0.5 sin x).
e^(−0.5x) ≠ 0, so cos x − 0.5 sin x = 0, which gives tan x = 2.
tan x > 0 in quadrants I and III: x = tan⁻¹ 2 ≈ 1.107 or x ≈ 1.107 + π ≈ 4.249.
y(1.107) ≈ 0.514 and y(4.249) ≈ −0.107.

Horizontal tangents at about (1.107, 0.514) and (4.249, −0.107). Each peak is lower than the one before: the wave is damped.

The damped wave y = e^(−0.5x) sin x. Its tangent at the origin is y = x, and its horizontal tangents are where tan x = 2, at x ≈ 1.107 and x ≈ 4.249.

Worked example

Example 4: Checking a derivative numerically

You found d/dx e^(sin x) = cos x e^(sin x). Check it at x = 1 with a calculator (radian mode), using h = 0.00001.

Centred difference: [e^(sin 1.00001) − e^(sin 0.99999)] ÷ 0.00002 ≈ 1.253381.
Your formula: cos 1 × e^(sin 1) ≈ 0.540302 × 2.319777 ≈ 1.253381.
They agree to six decimal places.

The derivative is confirmed. This is the same kind of verification with technology that the curriculum asks for when checking d/dx aˣ = aˣ ln a.

Worked example

Example 5: Error detective

A student wrote: d/dx (x² sin 3x) = 2x × 3 cos 3x = 6x cos 3x. Find the error and correct it.

The student multiplied the two derivatives. That is not the product rule.
Product rule: (x²)′ sin 3x + x² (sin 3x)′.
(sin 3x)′ = 3 cos 3x by the chain rule.
Correct answer: 2x sin 3x + 3x² cos 3x.

2x sin 3x + 3x² cos 3x. Quick numerical check at x = 1: the student's answer gives about −5.94, while the centred difference gives about −2.69, which matches the correct formula: 2 sin 3 + 3 cos 3 ≈ −2.69.

Hands-on activity

Derivative memory game

You need: 24 index cards (or squares of paper), a pencil, and a partner (a parent, sibling or friend; they don't need to know calculus).

On 12 cards, write these functions: e^(4x), 3ˣ, x eˣ, sin(2x), cos(x²), tan x, eˣ cos x, sin² x, e^(−x), x sin x, 2^(3x), sin x / x.
On the other 12 cards, write their derivatives (work them out first and check each one numerically at x = 1 with a centred difference).
Shuffle and lay all 24 cards face down. Take turns flipping two cards. If they are a function and its derivative, keep the pair. Your partner can use your answer key to check.
After the game, pick the three pairs you found hardest and redo them from scratch.

Answer key: 4e^(4x); 3ˣ ln 3; eˣ(1 + x); 2 cos(2x); −2x sin(x²); sec² x; eˣ(cos x − sin x); 2 sin x cos x; −e^(−x); sin x + x cos x; 3 ln 2 × 2^(3x); (x cos x − sin x)/x².

Just for fun

A calculus student's to-do list: 1. Differentiate eˣ. 2. Admire the answer. 3. Differentiate it again, just to be sure. 4. Realize this could go on forever.

Your turn

Differentiate y = x³ e^(−2x).

Differentiate y = cos(e^x).

Differentiate y = e^(x²) sin x.

Differentiate y = 2ˣ cos x and evaluate at x = π.

Find the equation of the tangent to y = e^(2x) cos x at x = 0.

Check numerically that d/dx (x sin x) at x = 2 is sin 2 + 2 cos 2.

More practice: warm-up

Which rule comes first for y = (x² + 1) eˣ?

Which rule comes first for y = e^(sin x)?

Differentiate y = 5eˣ − 3 sin x + x⁴.

Differentiate y = sin(πx).

What is the slope of y = eˣ + sin x at x = 0?

More practice: on your own

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

Find all x in [0, 2π] where y = eˣ sin x has a horizontal tangent.

Find the normal line to y = sin x at x = π/3.

A student wrote d/dx e^(3x²) = 3x² e^(3x² − 1). Explain the error and correct it.

Differentiate y = sin³(2x).

Challenge

Find the second derivative of y = e^(−x) cos x and show that y″ + 2y′ + 2y = 0.

(a) Find the point on y = eˣ − 2x where the tangent is horizontal. (b) Find where the tangent to y = sin x + eˣ at x = 0 crosses the x-axis.

Find the values of x in [0, 2π] where the tangents to y = sin x and y = cos x are parallel.

Check your understanding

Quiz

Which rule should you apply first to y = sin(x²) eˣ?

d/dx (x² e^(3x)) =

The tangent to y = e^(2x) cos x at x = 0 is:

A centred difference quotient for f′(a) is:

Where does y = e^(−0.5x) sin x have horizontal tangents?

A student wrote d/dx (x² sin 3x) = 6x cos 3x. The mistake was:

For parents and teachers

Big idea (A3.5, with A2.4 and A2.8): with all the rules in hand, the skill is reading a function's structure and applying the rules in the right order, then checking. A centred difference on a calculator is a reliable way to check any derivative.

How to help: ask "What would you do last if you were evaluating this on a calculator?" That operation decides the first rule. The memory game in the activity makes practice social; you can hold the answer key.

Common difficulties:

Multiplying derivatives instead of using the product rule.
Missing a layer of the chain rule in functions like sin(e^(2x)).
Treating eˣ like xⁿ.
Radian-mode slips when checking numerically.