Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
a) If \(y=\left(2x^{3}-x\right)^{5} \) find \(\frac{dy}{dx}\).
b) Find the gradient of the curve \(y=3x\cos\left(2x\right)\) at the point \(\left(\frac{5\pi}{6},\frac{5\pi}{4}\right)\).
c) Make \(\frac{dy}{dx}\) the subject of \(3\frac{dy}{dx}+x\frac{dy}{dx}=7\).
Solutions
Get Ready
Questions
Find the gradient of the circle \(x^{2}+y^{2}=25\) at the point where \(x=3\).
Solutions
Notes
| Explicitly Defined | Implicitly Defined |
|---|---|
| \( y = 3x^2 - 5x + 2 \) | \( x^2 + y^2 = 9 \) |
| \( y = \sqrt{x + 1} \) | \( y^2 - x = 1 \) |
| \( y = \frac{1}{x} \) | \( xy = 1 \) |
| \( y = e^{2x} \) | \( \ln(y) - 2x = 0 \) |
| \( y = \sin(x) \) | \( \sin(x+y) = y^2 \) |
So far we have only considered explicitly defined functions.
An implicitly defined function is one that is NOT stated in the form \(y=\)
Examples and Your Turns
Example
Differentiate with respect to \(x\) the function$$y^{2}=4x$$
Your Turn
Find the gradient of the circle \(x^{2}+y^{2}=25\) at the point \(\left(3,4\right)\).
Notes
In general$$\frac{d}{dx}\left(f\left(y\right)\right)=f’\left(y\right)\times \frac{dy}{dx}$$
That is, when you differentiate a term containing \(y\) you have to multiply by \(\frac{dy}{dx}\).
Quick Derivatives
For each of the following implicit terms, find the derivative.
These are the common terms that can form part of implicit functions, and knowing them is very helpful.
| Term | Derivative |
|---|---|
| \( y^2 \) |
\( 2y \frac{dy}{dx} \)
?
|
| \( y \) |
\( \frac{dy}{dx} \)
?
|
| \( y^3 \) |
\( 3y^2 \frac{dy}{dx} \)
?
|
| \( y^5 \) |
\( 5y^4 \frac{dy}{dx} \)
?
|
| \( xy \) |
\( x \frac{dy}{dx} + y \)
?
|
| \( x^2 y \) |
\( x^2 \frac{dy}{dx} + 2xy \)
?
|
| \( xy^2 \) |
\( 2xy \frac{dy}{dx} + y^2 \)
?
|
| \( x^2 y^2 \) |
\( 2x^2 y \frac{dy}{dx} + 2xy^2 \)
?
|
| \( \frac{x}{y} \) |
\( \frac{y - x \frac{dy}{dx}}{y^2} \)
?
|
| \( \frac{y}{x} \) |
\( \frac{x \frac{dy}{dx} - y}{x^2} \)
?
|
| \( \frac{x^2}{y} \) |
\( \frac{2xy - x^2 \frac{dy}{dx}}{y^2} \)
?
|
| \( \frac{y^2}{x} \) |
\( \frac{2xy \frac{dy}{dx} - y^2}{x^2} \)
?
|
Examples and Your Turns
Example
\(x^{3}+y^{3}-9xy\) is known as the folium of Descartes. Find the derivative function \(\frac{dy}{dx}\).
Your Turn
Find \(\frac{dy}{dx}\) for \(x^{2}y+y^{2}=10\).
Your Turn
Find the coordinates of the turning points on the curve \(y^{3}+3xy^{2}-x^{3}=27\).
Your Turn
A curve is defined by the equation \(2x^{2}-3xy+y^{2}=-8\).
a) Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\).
b) Find the coordinates of the points on the curve where the tangent is horizontal.
c) Find the coordinates of the points on the curve where the tangent is vertical.
Find $\frac{dy}{dx}$ using Implicit Differentiation
1. Original Equation
2. Differentiate w.r.t x (expanding any quotient rules)
3. Collect dy/dx terms on the left
4. Factor out dy/dx
5. Isolate dy/dx
Your Turn
Find \(\frac{dy}{dx}\) for $$\left(x+y\right)^{4}=2y^{2}$$
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
How is the idea of implicit differentiation linked to the Chain Rule? Why do we multiply by \(\frac{dy}{dx}\)?
Why is it better to differentiate \(x^{2}+y^{2}=25\) implicitly rather than rearrange to \(y=\sqrt{25-x^{2}}\) and differentiate the explicit function?
Common Mistakes / Misconceptions
Always remember to multiply derivatives of \(y\) by \(\frac{dy}{dx}\).
Don’t forget that you might have to apply the product or quotient rules to implicit functions. Remember that \(xy\) is a product, not a single term.
Watch out for constant terms that need to differentiate to \(0\).
Connecting This to Other Skills
You need to be able to Differentiate Polynomials (4.5), and apply the Product Rule (4.8) and Quotient Rule (4.9).
The whole basis of Implicit Differentiation is the Chain Rule (4.7).
You might need to find Tangents and Normals (4.13) or Stationary Points (4.14) on implicitly defined functions.
In the next skills we will see that we can also need to Differentiate Exponentials and Logs (4.11) and Differentiate Trig (4.12) in implicitly defined functions.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?