Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
Expand and differentiate \(y=\left(3x+10\right)^{3}\)
Solutions
Get Ready
Questions
Find \(g\left(x\right)\) and \(f\left(x\right)\) such that \(g\left(f\left(x\right)\right)\) is:
a) \(\left(3x+10\right)^{3}\)
b) \(\frac{1}{2x+4}\)
c) \(\sqrt{x^{2}−3x}\)
d) \(\frac{10}{\left(3x-x^{2}\right)^{3}}\)
e) \(\sin\left(5x−3\right)\)
f) \(e^{4x^{2}−3x}\)
g) \(\ln\left(1−2x\right)\)
h) \(\cos^{2}x\)
Solutions
Notes
The Chain Rule allows us to differentiate composite functions of the form \(y=g\left(f\left(x\right)\right)\).
It states that:
If \(y=g\left(u\right)\) where \(u=f\left(x\right)\) then $$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$$
Examples and Your Turns
Example
Find \(\frac{dy}{dx}\) for the function $$y=\left(3x+10\right)^{3}$$
Your Turn
Differentiate $$y=\frac{1}{2x+4}$$
Your Turn
Differentiate $$y=\sqrt{x^{2}−3x}$$
Your Turn
Differentiate $$y=\frac{10}{\left(3x−x^{2}\right)^{3}}$$
Exercise
Differentiate each of these functions:
a) \(y=\left(3x−7\right)^{5}\)
b) \(y=\frac{4}{\left(2x-3\right)^{7}}\)
c) \(y=\frac{1}{4−5x}\)
d) \(y=\frac{10}{x^{2}+4x}\)
e) \(y=\sqrt{5x−x^{3}}\)
f) \(y=3\left(x^{3}+\frac{1}{x}\right)^{4}\)
Differentiate the function using the Chain Rule
Notes - The Quick Way
Differentiate the ‘outer’ function by normal rules.
Multiply by the derivative of the inner function.
E.g. Differentiate \(y=\left(4x−3\right)^{5}\)
Your Turn
The derivative of \(f\left(x\right)=\left(3x−b\right)^{a}\) is \(f′\left(x\right)=108x^{2}−72x+12\).
Find the constants \(a\) and \(b\).
Your Turn
Suppose \(y=a\sqrt{1+bx}\) where \(a\) and \(b\) are constants.
When \(x=2\), \(y=2\) and \(\frac{dy}{dx}=−12\).
Find \(a\) and \(b\).
Your Turn
Suppose \(f\left(x\right)=2\left(ax−b^{2}\right)^{3}\). Given that \(f\left(1\right)=16\) and \(f′\left(1\right)=120\) find the positive values of \(a\) and \(b\).
Your Turn
Consider the function \(f\left(x\right)=\sqrt{ax+b}\) where \(a\) and \(b\) are constants. The graph of \(f\) passes through the point \(P\left(5,4\right)\). The gradient of the tangent to the curve at the point \(P\) is \(\frac{3}{8}\).
Find the value of \(a\) and \(b\).
Challenge
Can you find:
a) functions \(f\left(x\right)\) and \(g\left(x\right)\) so that \(f\left(g\left(x\right)\right)\) has stationary points when \(x=−1\) and \(x=5\)?
b) functions \(f\left(x\right)\) and \(g\left(x\right)\) so that \(f\left(g\left(x\right)\right)\) has a local minumum?
c) functions \(f\left(x\right)\) and \(g\left(x\right)\) so that \(f\left(g\left(x\right)\right)\) has no stationary points?
Adapted from https://undergroundmathematics.org/chain-rule/can-you-find-chain-rule-edition
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
Can you explain why the ‘quick method’ works based on the formal rule?
Why might the Chain Rule be preferable to expanding a binomial and the differentiating?
Common Mistakes / Misconceptions
The most common error in the chain rule is to miss the multiplication by the derivative of the inner function.
Another common error is to differentiate the inner function within the outer function. For example, in \(y=\left(2x^{2}+3x\right)^{4}\), getting \(\frac{dy}{dx}=4\times \left(4x+3\right)^{3}\).
On a similar note, if the inner derivative is an expression and not just a single term, it must go inside brackets to ensure the whole expression is being multiplied.
Connecting This to Other Skills
We are using the skills of Differentiating Polynomials (4.5) and connecting this to the idea of Composite Functions (2.12).
We will see how the Chain Rule will be incorporated into the Product Rule (4.8) and Quotient Rule (4.9).
It is also the basis of Implicit Differentiation (4.10).
When we learn to Differentiate Exponentials and Logarithms (4.11) and Differentiate Trig (4.12) we will use the Chain Rule.
And the application of finding Related Rates of Change (4.19) uses the Chain Rule.
We will learn the Reverse Chain Rule (5.4) when studying integration.
The Chain Rule is one of the most common topics to appear in exams as it appears in the majority of questions involving calculus, especially at higher level.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?