Welcome to the Next Level of Differentiation!
In your Pure Mathematics 1 (P1) journey, you learned how to differentiate basic powers of \(x\) like \(x^2\) or \(3x^5\). In Pure Mathematics 2 (P2), we are going to expand your "toolkit." We will learn how to handle more interesting functions like exponentials, logarithms, and trigonometry, and learn some clever rules for when functions are multiplied or divided.
Don’t worry if this seems like a lot of new symbols. Think of it like learning new moves in a video game—once you know the button combinations, you can tackle much bigger bosses!
1. The New "Core" Functions
In P2, you need to memorize five new derivatives. These are your building blocks.
Exponentials and Logarithms
1. The Exponential Function: \( \frac{d}{dx}(e^x) = e^x \)
Analogy: The function \(e^x\) is the "coolest" function in math. It is so happy where it is that when you differentiate it, it stays exactly the same!
2. The Natural Logarithm: \( \frac{d}{dx}(\ln x) = \frac{1}{x} \)
Trigonometric Functions
3. Sine: \( \frac{d}{dx}(\sin x) = \cos x \)
4. Cosine: \( \frac{d}{dx}(\cos x) = -\sin x \)
5. Tangent: \( \frac{d}{dx}(\tan x) = \sec^2 x \)
Quick Review Box:
- \(e^x \rightarrow e^x\)
- \(\ln x \rightarrow \frac{1}{x}\)
- \(\sin x \rightarrow \cos x\)
- \(\cos x \rightarrow -\sin x\) (Watch that minus sign!)
- \(\tan x \rightarrow \sec^2 x\)
Common Mistake: Students often forget the negative sign when differentiating \(\cos x\). A simple way to remember: If the original trig function starts with a "c" (like cosine), its derivative is negative.
2. The Chain Rule (Refresher for P2)
When you have a "function inside a function" (a composite function), you must use the Chain Rule. You differentiate the "outside" and then multiply by the derivative of the "inside."
Example: Differentiate \(y = e^{3x+2}\)
- Outside: \(e^{(\dots)}\) stays as \(e^{(\dots)}\).
- Inside: The derivative of \(3x+2\) is \(3\).
- Result: \(\frac{dy}{dx} = 3e^{3x+2}\)
Key Takeaway:
Always check if there is a "nested" function. If the \(x\) in your trig or log function is actually a more complex term (like \(5x\) or \(x^2\)), you must multiply by its derivative at the end.
3. The Product Rule
Use this rule when you have two different functions multiplied together, such as \(x^2 \sin x\).
If \(y = uv\), then:
\(\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}\)
Memory Aid: "Left D-Right plus Right D-Left."
(The first part stays same \(\times\) derivative of the second) + (The second part stays same \(\times\) derivative of the first).
Example: Differentiate \(y = x^2 \ln x\)
- Let \(u = x^2\) and \(v = \ln x\)
- \(\frac{du}{dx} = 2x\) and \(\frac{dv}{dx} = \frac{1}{x}\)
- \(\frac{dy}{dx} = (x^2)(\frac{1}{x}) + (\ln x)(2x)\)
- Simplified: \(\frac{dy}{dx} = x + 2x \ln x\)
4. The Quotient Rule
Use this rule when one function is divided by another, like \(\frac{\sin x}{e^x}\).
If \(y = \frac{u}{v}\), then:
\(\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\)
Memory Aid: "Low D-High minus High D-Low, over Low-Low."
- Low: bottom function (\(v\))
- High: top function (\(u\))
- D: derivative
- Low-Low: the bottom squared (\(v^2\))
Key Takeaway: The order of the top part (the numerator) matters! You must start with the bottom function (\(v\)) times the derivative of the top.
5. Implicit Differentiation
Usually, we see equations like \(y = \dots\). But what if \(x\) and \(y\) are all mixed up, like \(x^2 + y^2 = xy + 7\)? This is an implicit function.
How to do it:
1. Differentiate everything with respect to \(x\) as usual.
2. Whenever you differentiate a term with a \(y\), simply do it normally but attach a \(\frac{dy}{dx}\) to the end of it.
3. Rearrange the equation to solve for \(\frac{dy}{dx}\).
Example: Differentiate \(y^2\)
Treat it like \((\dots)^2\). The derivative is \(2y \times \frac{dy}{dx}\).
Did you know? This technique is incredibly useful for finding the gradient of circles or ovals where you can't easily write "\(y = \dots\)"!
6. Parametric Differentiation
Sometimes \(x\) and \(y\) are both defined by a third variable, often \(t\) (time) or \(\theta\) (angle). These are called parameters.
Example: \(x = 2t\), \(y = t^2\)
To find \(\frac{dy}{dx}\), use this simple formula:
\(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\)
Step-by-step:
1. Find \(\frac{dx}{dt}\).
2. Find \(\frac{dy}{dt}\).
3. Divide the \(y\)-result by the \(x\)-result.
7. Applications: Tangents and Normals
This is where we use our derivatives to solve geometry problems. This is exactly like P1, but with our new P2 functions!
- Tangent Gradient: This is just the value of \(\frac{dy}{dx}\) at your specific point.
- Normal Gradient: This is the "negative reciprocal" of the tangent gradient. (Flip it and change the sign: \(-\frac{1}{m}\)).
- Equation: Use the straight-line formula: \(y - y_1 = m(x - x_1)\).
Quick Review Box:
- Product Rule: For \(u \times v\)
- Quotient Rule: For \(u \div v\)
- Implicit: For mixed \(x\) and \(y\). (Add \(\frac{dy}{dx}\) for \(y\) terms!)
- Parametric: For when \(x\) and \(y\) depend on \(t\).
Don't worry if this seems tricky at first! Differentiation is all about practice. Once you learn to recognize which "rule" to use (is it a product? is it a quotient? is it implicit?), the rest is just following the steps you've memorized. You've got this!