Trig Derivative Worksheet Trig Derivative Duel
Calculus Unit 3: Transcendental Functions
NAME: ___________________________
DATE: ___________________________
\[ \frac{d}{dx}\sin(x)=\cos(x) \]
\[ \frac{d}{dx}\cos(x)=-\sin(x) \]
\[ \frac{d}{dx}\tan(x)=\sec^2(x) \]
\[ \frac{d}{dx}\cot(x)=-\csc^2(x) \]
\[ \frac{d}{dx}\sec(x)=\sec(x)\tan(x) \]
\[ \frac{d}{dx}\csc(x)=-\csc(x)\cot(x) \]
01
Find \( \frac{dy}{dx} \) for the function: \[ y=x^2\sin(x) \]
02
Find the derivative of the quotient: \[ f(x)=\frac{\tan(x)}{e^x} \]
03
Calculate the derivative \( g'(x) \): \[ g(x)=\cos(x^3+4x) \]
04
Evaluate at the point \( x=\pi \): \[ h(x)=\sec(x)+\csc(x) \]
05
Differentiate using the chain rule: \[ y=\sin^2(5x) \]
06
Ultimate Challenge
Find \( \frac{dy}{dx} \) using the quotient and chain rules: \[ y=\frac{\sin(x)\cos(x)}{1+\tan(x)} \]
© MIXED LEARNING LAB CALCULUS SERIES
REFER TO TEACHER GUIDE FOR SOLUTIONS
Trig Derivative Slides Trig Derivative Duel
Mastering the Calculus of Oscillation
CALCULUS I UNIT 3.4
The Core Six Derivatives
d/dx sin(x) = cos(x)
d/dx cos(x) = -sin(x)
d/dx tan(x) = sec²(x)
d/dx cot(x) = -csc²(x)
d/dx sec(x) = sec(x)tan(x)
d/dx csc(x) = -csc(x)cot(x)
Pro Tip: All trig functions starting with "co-" have negative derivatives.
The Power of the Chain
Differentiate:
y = sin(x²)
Step-by-Step
Identify inner: u = x²
Identify outer: f(u) = sin(u)
Apply Chain Rule: f'(u) · u'
Combine: cos(x²) · (2x)
Final Result:
y' = 2x cos(x²)
Rapid Fire: Product Rule
Quick! What is the derivative of:
f(x) = x · tan(x)
A: x · sec²(x)
B: tan(x) + x · sec²(x)
C: 1 + sec²(x)
D: x · tan(x) + sec²(x)
Trig Derivative Key Teacher Guide Trig Derivative Duel: KEY
Teacher Reference & Solutions
Key Version 1.2
01. \( y = x^2 \sin(x) \)
Product Rule: \( \frac{d}{dx}[uv] = u'v + uv' \)
\( y' = \displaystyle (2x)(\sin x) + (x^2)(\cos x) = 2x \sin x + x^2 \cos x \)
02. \( f(x) = \frac{\tan x}{e^x} \)
Quotient Rule: \( \displaystyle \frac{u'v - uv'}{v^2} \)
\( f'(x) = \displaystyle \frac{\sec^2(x)e^x - \tan(x)e^x}{(e^x)^2} = \frac{\sec^2 x - \tan x}{e^x} \)
03. \( g(x) = \cos(x^3 + 4x) \)
Chain Rule: \( f'(g(x)) \cdot g'(x) \)
\( g'(x) = \displaystyle -\sin(x^3 + 4x) \cdot (3x^2 + 4) = -(3x^2 + 4)\sin(x^3 + 4x) \)
04. \( h(x) = \sec(x) + \csc(x) \) at \( x = \pi \)
\( h'(x) = \displaystyle \sec x \tan x - \csc x \cot x \)
05. \( y = \sin^2(5x) \)
\( y' = \displaystyle 2\sin(5x) \cdot \cos(5x) \cdot 5 = 10\sin(5x)\cos(5x) = 5\sin(10x) \)
06. Challenge: \( y = \frac{\sin x \cos x}{1 + \tan x} \)
Numerator: \( u = \sin x \cos x \rightarrow u' = \cos^2 x - \sin^2 x = \cos(2x) \)
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