Lesson 4: Trig, Exp & Log Derivatives

Lesson at a Glance | He Tirohanga Whakamua

Do NowWhy calculus angles MUST be in Radians and why d/dx(e^x) = e^x10 min

Trig DerivativesDifferentiating

\sin (u), \cos (u), \tan (u), \sec (u)

with chain rule15 min

Exp & Log DerivativesDifferentiating

e^{g (x)}

and

\ln (g (x)) \to \frac{g^{'} (x)}{g (x)}

15 min

Portfolio EntryWrite Section 4: Transcendental Function Derivatives Guide10 min

Exit DrillDifferentiate

y = e^{4 x} \cos (3 x)

using product rule5 min

Ngā Whāinga Ako | Learning Intentions

Students will know

Trigonometric derivatives (angles in radians!):
• $\frac{d}{d x} [\sin (u)] = \cos (u) \cdot u^{'}$
• $\frac{d}{d x} [\cos (u)] = - \sin (u) \cdot u^{'}$
• $\frac{d}{d x} [\tan (u)] = \sec^{2} (u) \cdot u^{'}$
Exponential & Logarithmic derivatives:
• $\frac{d}{d x} [e^{u}] = e^{u} \cdot u^{'}$
• $\frac{d}{d x} [\ln (u)] = \frac{u^{'}}{u}$

Students will demonstrate

By executing product and quotient derivatives containing combinations of $e^{x}, \ln x$ , and trigonometric functions.
By completing Section 4 of their Level 3 Calculus Differentiation Portfolio.

Do Now | Tīmatanga Whakaaro (10 min)

Natural Constant Prompt:

"Why is Euler's number $e \approx 2.71828$ so famous in calculus? What makes $f (x) = e^{x}$ the only function whose derivative is identical to itself?"

Unpack: Because $e$ is defined as the unique base where the slope of the tangent line at $(0, 1)$ is exactly equal to $1$ ( ${lim}_{h \to 0} \frac{e^{h} - 1}{h} = 1$ )! Therefore, $\frac{d}{d x} [e^{x}] = e^{x}$ .

Transcendental Derivative Formulas (15 min)

1. Exponential Product Example

$y = e^{4 x} \cos (3 x)$
$u^{'} = 4 e^{4 x}, v^{'} = - 3 \sin (3 x)$
$\frac{d y}{d x} = 4 e^{4 x} \cos (3 x) - 3 e^{4 x} \sin (3 x) = e^{4 x} [4 \cos (3 x) - 3 \sin (3 x)]$ .

2. Natural Log Chain Example

$y = \ln (5 x^{3} + 2 x)$
$\frac{d y}{d x} = \frac{1}{5 x^{3} + 2 x} \cdot (15 x^{2} + 2) = \frac{15 x^{2} + 2}{5 x^{3} + 2 x}$ .

📁 Calculus Differentiation Portfolio — Section 4: Trig, Exp & Log Derivatives

Students open their Level 3 Calculus Portfolio and complete Section 4:

Section 4 Requirements:

1. Transcendental Derivative Summary Table: Write out formulas for $\sin u, \cos u, \tan u, \sec u, e^{u}, \ln u$ , highlighting inner chain derivative $u^{'}$ .

2. Mixed Derivative Solver: Differentiate $y = \frac{\ln (3 x)}{x^{2}}$ and $y = \tan^{3} (4 x) = (\tan (4 x))^{3}$ .

3. Radians Rationale: 1-paragraph explanation of why the standard trigonometric derivative formulas assume angles are measured in radians. When degrees are used, conversion to radians introduces a factor of $π / 180$ .

Kaiako answer and checking guide

Scope: Mathematical checking support for the shipped tasks. This is not an NZQA marking schedule or an assessment-evidence requirement.

Exit drill: dy/dx = e^4x[4 cos(3x) − 3 sin(3x)].
Portfolio 1: For an inner function u, the six derivatives are cos(u)u′, −sin(u)u′, sec²(u)u′, sec(u)tan(u)u′, e^uu′, and u′/u.
Portfolio 2: The derivatives are [1 − 2 ln(3x)]/x³, for x > 0, and 12 tan²(4x)sec²(4x).
Portfolio 3: In radians, lim_x→0 sin(x)/x = 1. In degrees, sin(πx/180) differentiates to (π/180)cos(πx/180), so calculus still works but the conversion factor remains.

Exit Verification | Ka Mutu Hoki (5 min)

Exit Check:

"My Section 4 differentiates y = e^{4x} cos(3x) using product rule yielding dy/dx = e^{4x}(4 cos(3x) - 3 sin(3x))."

Teacher Planning & NCEA Alignment

NCEA Level 3 Calculus Alignment (6 Credits External):

Differentiation Methods: Differentiate trigonometric ( $\sin, \cos, \tan, \sec$ ), exponential ( $e^{u}$ ), and logarithmic ( $\ln u$ ) functions.

Vocabulary: Transcendental function, natural logarithm ( $\ln$ ), exponential base ( $e$ ), radian measure, chain rule, product rule.

Lesson 4: Trigonometric, Exponential & Logarithmic Derivatives