Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
a) Simplify \(e^{\ln\left(5x\right)}\) and \(\ln\left(e^{3x}\right)\).
b) Write \(\ln\left(x^{2}\sqrt{y}\right)\) in the form \(a\ln x +b\ln y\) where \(a,b\) are rational constants to be determined.
c) If \(y=f\left(g\left(x\right)\right)\) then what is \(\frac{dy}{dx}\)?
Solutions
Get Ready
Questions
Shown below is the graph of \(y=2^{x}\). On the graph sketch the gradient function.
Reveal the gradient function by clicking on the circle on line 3. Does it match what you drew?
What do you notice about the shape of the gradient function compared to the original function?
Change the value ‘2’ to ‘3’ so now you have the graph of \(y=3^{x}\) and its derivative function?
What is different between the relationship between \(y=2^{x}\) and \(y=3^{x}\) with their respective gradient functions?
This means that there must be some value between \(a=2\) and \(a=3\) where the gradient function of \(y=a^{x}\) is exactly the same as the original function.
Use the slider to adjust the value of \(a\) to determine when this happens.
What is this value to 3 decimal places?
What do you notice about this value?
Solutions
Notes
The derivative of any exponential function is an exponential function.
The function \(y=e^{x}\) is its own derivative.
| Function | Derivative |
|---|---|
| \( e^x \) |
\( e^x \)
?
|
| \( e^{f(x)} \) |
\( f'(x)e^{f(x)} \)
?
|
Examples and Your Turns
Example
Find the derivative of the function$$f\left(x\right)=e^{2x}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=e^{4x^{2}-3x}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=2e^{x}+e^{-3x}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=xe^{x}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=e^{e^{2x}}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=x^{2}e^{-3x}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=\frac{e^{2x}}{x}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=\frac{x}{e^{2x}}$$
Your Turn
Given that \(f\left(x\right)=e^{kx}+x\) and \(f’\left(0\right)=-8\) find \(k\).
Your Turn
Given that \(f\left(x\right)=e^{kx}+2x^{2}\) and \(f’\left(0\right)=-5\) find the value of \(k\).
Investigation
What type of function is shown in the graph below?
Sketch the gradient function, and the reveal it by clicking the empty circle. Did you get it right?
What type of function is the derivative?
Notes
The derivative of any logarithmic function is a rational function.
The function \(y=\ln x\) is particularly important.
| Function | Derivative |
|---|---|
| \( \ln x \) |
\( \frac{1}{x} \)
?
|
| \( \ln\left(f(x)\right) \) |
\( \frac{f'(x)}{f(x)} \)
?
|
Examples and Your Turns
Example
Find the derivative of the function$$f\left(x\right)=\ln 5x$$
Your Turn
Find the derivative of the function$$f\left(x\right)=5\ln x$$
Your Turn
Find the derivative of the function$$f\left(x\right)=\ln \left(1-3x\right)$$
Your Turn
Find the derivative of the function$$f\left(x\right)=\ln \left(6x-1\right)$$
Your Turn
Find the derivative of the function$$f\left(x\right)=x^{3}\ln x$$
Your Turn
Find the derivative of the function$$f\left(x\right)=e^{2x}\ln x^{2}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=\frac{\ln 5x}{x}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=\frac{\ln x^{3}}{e^{x}}$$
Your Turn
Find the derivative of the function$$f\left(x\right)=\ln \left(xe^{-x}\right)$$
Your Turn
Find the derivative of the function$$f\left(x\right)=\ln\left(\frac{x^{2}}{\left(x+2\right)\left(x-3\right)}\right)$$
Notes (HL)
In the HL course you also have to be able to use the derivatives of other exponential and logarithmic functions:
| Function | Derivative |
|---|---|
| \( a^{x} \) |
\( \left(\ln a\right)\times a^{x} \)
?
|
| \( \log_{a} x \) |
\( \frac{1}{x\ln a} \)
?
|
Examples and Your Turns (HL)
Example
Find the derivative of the function$$y=4^{x}$$
Your Turn
Find the derivative of the function$$y=4^{2x}$$
Example
Find the derivative of the function$$y=\log_{2} x$$
Your Turn
Find the derivative of the function$$y=\log_{8} x$$
Your Turn
Find the derivative of the function$$y=x4^{x}$$
Your Turn
Find the derivative of the function$$y=x\log_{3} x$$
Notes (HL)
We can also apply the ideas of implicit differentiation to exponentials and logarithms.
Your Turn
Find the derivative of the function$$x^{3}e^{3y}+4x^{2}y^{3}=27e^{-2x}$$
Your Turn
A curve is defined by the equation \(e^{x-y}=x^{2}-y+k\) where \(k\) is a constant.
The curve passes through the point \(P\left(2,2\right)\).
a) Find the value of \(k\).
b) Find the gradient of the tangent to the curve at \(P\).
Your Turn
The equation of a curve is given by \(a\ln\left(y\right)+x^{2}y=b\) where \(a\) and \(b\) are non-zero constants.
It is known that the point \(Q\left(1,e\right)\) lies on the curve and that the gradient of the tangent at \(Q\) is \(-\frac{2e}{1+e^{2}}\).
Find the exact values of \(a\) and \(b\).
Notes
You are not required to know these proofs, but they are provided for completeness.
Theorem
$$e=\lim_{t\to 0} \left(1+t\right)^{\frac{1}{t}}$$
-
Recall from 1.10 Natural Exponential that $$e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}$$Let \(n=\frac{1}{t}\) and we get$$e=\lim_{t\to 0}\left(1+t\right)^{\frac{1}{t}}$$
Theorem
$$\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$$
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By 4.4 First Principles$$\begin{align}\frac{d}{dx}\left(\ln x\right)&=\lim_{h\to 0}\frac{\ln\left(x+h\right)-\ln\left(x\right)}{h}\\&=\lim_{h\to 0}\frac{\ln\left(\frac{x+h}{x}\right)}{h}\\&=\lim_{h\to 0} \frac{1}{h}\ln\left(1+\frac{h}{x}\right)\end{align}$$
Let \(t=\frac{h}{x}\implies h=xt\) which gives:$$\begin{align}\lim_{h\to 0} \frac{1}{h}\ln\left(1+\frac{xt}{x}\right)&=\lim_{t\to 0} \frac{1}{xt}\ln\left(1+t\right)\\&=\lim_{t\to 0} \frac{1}{x}\ln\left(1+t\right)^{\frac{1}{t}}\\&=\frac{1}{x}\lim_{t\to 0} \ln\left(1+t\right)^{\frac{1}{t}}\\&=\frac{1}{x}\ln e\\&=\frac{1}{x}\end{align}$$
Theorem
$$\frac{d}{dx}\left(e^{x}\right)=e^{x}$$
-
Let \(y=e^{x}\) which means that $$\ln y=x$$Differentiating implicitly we get$$\frac{1}{y}\frac{dy}{dx}=1$$So $$\begin{align}\frac{dy}{dx}&=y\\&=e^{x}\end{align}$$
Theorem
$$\frac{d}{dx}\left(a^{x}\right)=\left(\ln a\right)a^{x}$$
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Let \(y=a^{x}\) which means that $$\ln y=x\ln a$$Differentiating implicitly we get$$\frac{1}{y}\frac{dy}{dx}=\ln a$$So $$\begin{align}\frac{dy}{dx}&=\left(\ln a\right)y\\&=\left(\ln a\right)a^{x}\end{align}$$
Theorem
$$\frac{d}{dx}\left(\log_{a} x\right)=\frac{1}{x\ln a}$$
-
We can use the 1.11 Change of Base rule to get $$y=\log_{a} x=\frac{\ln x}{\ln a}$$ which differentiates as $$\frac{dy}{dx}=\frac{1}{\ln a}\times\frac{1}{x}=\frac{1}{x\ln a}$$
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
Why is the derivative of \(\ln\left(2x\right)\) the same as the derivative of \(\ln\left(x\right)\)?
Is it easier to differentiate logarithms using the chain rule or using laws of logarithms first?
Common Mistakes / Misconceptions
The derivative of \(\ln f\left(x\right)\) is NOT \(\frac{1}{f’\left(x\right)}\).
When differentiating \(e^{f\left(x\right)}\) don’t forget to multiply by the original \(e^{f\left(x\right)}\).
Connecting This to Other Skills
An understanding of what Logarithms (1.1) are and the Laws of Logarithms (1.2) is essential here. Similarly, you need to know the Natural Exponential \(e\) (1.10).
When solving problems, you might be required to solve Equations with Exponentials and Logarithms (1.12).
You need to Interpret Derivatives (4.6) and be able to use the Chain Rule (4.7), Product Rule (4.8) and Quotient Rule (4.9) as well as use Implicit Differentiation (4.10).
The rules, especially the derivative of \(\ln f\left(x\right)\), are of vital important in the Reverse Chain Rule (5.4).
Exponential Models often form the basis of Differential Equations (8.4) and in particular the population models that are Separable Differential Equations (8.6).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?