Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
a) If \(y=\left(f\left(x\right)\right)^{3} \) find \(\frac{dy}{dx}\) in terms of \(f\left(x\right)\) and \(f’\left(x\right)\)
b) Differentiate \(y=x^{2}e^{3x}\)
c) Find \(f’\left(x\right)\) when \(f\left(x\right)=\frac{4x-1}{x^{2}}\)
d) Rewrite \(\frac{1}{\cos ^{2} x}\)
e) Simplify \(\frac{\sin x}{\cos x}\) and \(1-\sin^{2} x\)
Solutions
Get Ready
Questions
What is the function shown in the graph below?
Sketch the derivative of this function, then reveal the answer above.
What function is the derivative?
Repeat for the following function.
In each graph, click on the settings spanner in the top right, and change to degrees.
Does the result you have found still hold? Why?
Notes
***IMPORTANT***
All calculus results involving trigonometry only work when you are in RADIANS.
You must always ensure you are working in radians when answering calculus questions with trigonometric functions.
To understand why, you can see the proofs below.
| Function | Derivative |
|---|---|
| \( \sin x \) |
\( \cos x \)
?
|
| \( \sin f(x) \) |
\( f'(x)\cos f(x) \)
?
|
| \( \cos x \) |
\( -\sin x \)
?
|
| \( \cos f(x) \) |
\( -f'(x)\sin f(x) \)
?
|
Examples and Your Turns
Example
Find the derivative of the function $$f\left(x\right)=\sin 3x$$
Your Turn
Find the derivative of the function $$f\left(x\right)=\cos \left(4x-3\right)$$
Your Turn
Find the derivative of the function $$f\left(x\right)=\sin ^{2} x$$
Your Turn
Find the derivative of the function $$f\left(x\right)=\cos ^{3} x$$
Your Turn
Find the derivative of the function $$f\left(x\right)=x\sin x$$
Your Turn
Find the derivative of the function $$f\left(x\right)=4\cos ^{2} 3x$$
Your Turn
Find the derivative of the function $$f\left(x\right)=e^{x}\sin x$$
Your Turn
Find the derivative of the function $$f\left(x\right)=\frac{\ln x}{\cos x}$$
Your Turn
Find the derivative of the function $$y=\sin^{2} 2x$$
Your Turn
Find the derivative of the function $$y=\sin x \cos x$$
Your Turn
Given \(f\left(x\right)=\sin \left(kx\right)+\cos x\) and \(f’\left(\frac{\pi}{2}\right)=3\) find the value of \(k\), where \(k>0\).
Your Turn
The curve \(y=a\sin x +b\cos x\) passes through \(\left(\frac{\pi}{4},\sqrt{2}\right)\) and has gradient \(2\) at that point.
Find \(a\) and \(b\).
Practice
Differentiate each of these.
| Function (sin) | Derivative | Function (cos) | Derivative |
|---|---|---|---|
| \( 2\sin x \) |
\( 2\cos x \)
?
|
\( 3\cos x \) |
\( -3\sin x \)
?
|
| \( \sin 2x \) |
\( 2\cos 2x \)
?
|
\( \cos 3x \) |
\( -3\sin 3x \)
?
|
| \( \sin^2 x \) |
\( 2\sin x \cos x \)
?
|
\( \cos^3 x \) |
\( -3\cos^2 x \sin x \)
?
|
| \( 2\sin^2 x \) |
\( 4\sin x \cos x \)
?
|
\( 3\cos^3 x \) |
\( -9\cos^2 x \sin x \)
?
|
| \( \sin^2 2x \) |
\( 4\sin 2x \cos 2x \)
?
|
\( \cos^3 3x \) |
\( -9\cos^2 3x \sin 3x \)
?
|
| \( 2\sin^2 2x \) |
\( 8\sin 2x \cos 2x \)
?
|
\( 3\cos^3 3x \) |
\( -27\cos^2 3x \sin 3x \)
?
|
Can you generalise any of the results?
The following proofs are not required for the course, but are provided for completeness.
Theorem
If \(\theta\) is in radians then $$\lim_{\theta\to 0} \frac{\sin \theta}{\theta}=1$$
Theorem
If \(\theta\) is in radians then $$\lim_{\theta\to 0} \frac{\cos \theta -1}{\theta}=0$$
Theorem
If \(\theta\) is in radians then $$\frac{d}{dx}\left(\sin x\right)=\cos x$$
Theorem
If \(\theta\) is in radians then $$\frac{d}{dx}\left(\cos x\right)=-\sin x$$
Notes (HL)
We can also differentiate the \(\tan\) function.
| Function | Derivative |
|---|---|
| \( \tan x \) |
\( \sec^{2} x \)
?
|
| \( \tan f(x) \) |
\( f'(x)\sec^{2} f(x) \)
?
|
Examples and Your Turns (HL)
Example
Find the derivative of the function \(f\left(x\right)=\tan 3x\)
Your Turn
Find the derivative of the function \(f\left(x\right)=\tan \left(2-3x\right)\)
Your Turn
Find the derivative of the function \(f\left(x\right)=\tan^{2} x\)
Your Turn
Find the derivative of the function \(f\left(x\right)=e^{2x}\tan x\)
Theorem
If \(\theta\) is in radians then $$\frac{d}{dx}\left(\tan x\right)=\sec^{2} x$$
Notes (HL)
We can also apply the ideas of implicit differentiation.
Your Turn
Find the derivative of the function \(\cos\left(x+y\right)=y^{2}+x\)
Your Turn
Find the derivative of the function \(e^{x}+x\sin y =\cos 2y\)
Your Turn
Find the derivative of the function \(\sin y=x^{2}\cos x\)
Your Turn
Find the derivative of the function \(\cos xy =y^{2}\sin x\)
Your Turn
A curve is defined by the equation \(\cos\left(xy\right)+y^{2}=k\) where \(k\) is a constant.
The curve passes through the point \(P\left(0,2\right)\).
a) Find the value of \(k\).
b) Find an expression for \(\frac{dy}{dx}\).
c) Show that the gradient of the tangent to the curve at \(P\) is \(0\).
Investigation (HL)
Differentiate \(y=\operatorname{cosec}x\) with respect to \(x\) using:
a) the Quotient Rule
b) the Chain Rule
Differentiate \(y=\sec x\) with respect to \(x\).
Differentiate \(y=\cot x\) with respect to \(x\).
Notes (HL)
| Function | Derivative |
|---|---|
| \( \operatorname{cosec} x \) |
\( -\operatorname{cosec} x\cot x \)
?
|
| \( \sec x \) |
\( \sec x\tan x \)
?
|
| \( \cot x \) |
\( -\operatorname{cosec}^{2} x \)
?
|
Your Turn
Find \(\frac{dy}{dx}\) for \(y=\operatorname{cosec} 3x\)
Your Turn
Find \(\frac{dy}{dx}\) for \(y=\sqrt{\cot \frac{x}{2}}\)
Example
If \(y=\arcsin x\) show that$$\frac{dy}{dx}=\frac{1}{\sqrt{1-x^{2}}}$$
Your Turn
If \(y=\arccos x\) show that$$\frac{dy}{dx}=-\frac{1}{\sqrt{1-x^{2}}}$$
Your Turn
If \(y=\arctan x\) show that$$\frac{dy}{dx}=\frac{1}{1+x^{2}}$$
Notes (HL)
| Function | Derivative |
|---|---|
| \( \arcsin x \) |
$$ \frac{1}{\sqrt{1-x^{2}}} $$
?
|
| \( \arccos x \) |
$$ -\frac{1}{\sqrt{1-x^{2}}} $$
?
|
| \( \arctan x \) |
$$ \frac{1}{1+x^{2}} $$
?
|
Your Turn
Differentiate$$y=\arctan 4x$$
Your Turn
Differentiate $$y=\arccos \sqrt{x-3}$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Graphically, why are \(\sin\) and \(\cos\) related through derivatives?
Why do the results only hold for Radians and not degrees?
Common Mistakes / Misconceptions
The most common mistake when differentiating with trigonometric functions is to be working in degrees. You MUST be in radians for all questions involving trigonometry and calculus.
All the derivatives you need are in the formula booklet, so don’t try to recall one and get it wrong - always check!
Misapplying the Chain Rule or Product Rule is very common. Use the full layout if in doubt.
Connecting This to Other Skills
This topic builds on the ideas already encountered in Unit 4, particularly the Chain Rule (4.7) and Product Rule (4.8), along with Implicit Differentiation (4.10). You also need knowledge of the Trigonometric Functions (3.9), Reciprocal Trig Functions (3.11) and Inverse Trig Functions (3.15).
We will see how to reverse this process in the Reverse Chain Rule (5.4) and Integrating More Trig Functions (5.6) which can be substantially more difficult.
We will see how to find the Maclaurin Series (8.2) for Trigonometric Functions, as well as solve Differential Equations (8.4) involving them.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?