Differentiation of standard functions

Introduction to Differentiating Standard Functions

Welcome! In this chapter, we are going to learn how to "find the gradient" of various types of mathematical curves. In your earlier studies, you might have differentiated simple polynomials. Now, we are going to expand your toolkit to include exponentials, logarithms, and trigonometry.

Differentiation is essentially the math of change. Whether it’s how fast a rocket accelerates or how a population grows, these "standard functions" are the building blocks for describing the world around us. Don't worry if it seems like a lot to memorize at first—with a few simple patterns, you'll be differentiating like a pro in no time!

1. The Power Rule (Revision and Expansion)

The Power Rule is the most fundamental tool in differentiation. It allows us to differentiate functions of the form \( f(x) = x^n \).

The Rule: If \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \).
In simple terms: Multiply by the power, then subtract one from the power.

In A Level Mathematics (H240), you are expected to use this for rational values of \( n \). This includes fractions and negative numbers.

Example 1 (Fractions): If \( y = \sqrt{x} \), first rewrite it as \( y = x^{1/2} \).
Differentiating gives: \( \frac{dy}{dx} = \frac{1}{2}x^{-1/2} \) or \( \frac{1}{2\sqrt{x}} \).
Example 2 (Negative indices): If \( y = \frac{1}{x^2} \), rewrite it as \( y = x^{-2} \).
Differentiating gives: \( \frac{dy}{dx} = -2x^{-3} \) or \( -\frac{2}{x^3} \).

Quick Review: The Power Rule Process

1. Rewrite any roots as fractional powers (e.g., \( \sqrt[3]{x} = x^{1/3} \)).
2. Move any \( x \) terms from the denominator to the top using negative powers.
3. Apply the rule: Bring the power down, reduce the power by 1.

Key Takeaway: Always "prepare" your algebra before you differentiate! Rewriting fractions and surds as powers makes the process much easier.

2. Exponential Functions: \( e^{kx} \) and \( a^{kx} \)

Exponential functions describe things that grow or decay rapidly. The most famous is \( e^x \), where \( e \) is Euler’s number (approx. 2.718).

Differentiating \( e^{kx} \)

The function \( e^x \) is unique because it is its own derivative! However, if there is a constant \( k \) in front of the \( x \), it "pops out" to the front.

The Rule: If \( y = e^{kx} \), then \( \frac{dy}{dx} = ke^{kx} \).

Example: If \( y = e^{5x} \), then \( \frac{dy}{dx} = 5e^{5x} \).
Example: If \( y = e^{-x} \), then \( \frac{dy}{dx} = -e^{-x} \).

Differentiating \( a^{kx} \)

Sometimes the base isn't \( e \). It might be a number like 2 or 10. For these, we have to include a natural log (\( \ln \)) term in our answer.

The Rule: If \( y = a^{kx} \), then \( \frac{dy}{dx} = k(a^{kx})\ln(a) \).

Example: If \( y = 2^{3x} \), then \( \frac{dy}{dx} = 3(2^{3x})\ln(2) \).

Did you know? The reason \( e^x \) is so popular in science is specifically because its gradient is equal to its value. It’s the only function (other than zero) that behaves this way!

Key Takeaway: When differentiating \( e^{kx} \), the exponential part never changes—you just multiply the whole thing by the coefficient of \( x \).

3. Logarithmic Functions: \( \ln x \)

The natural logarithm, \( \ln x \), is the inverse of the exponential function \( e^x \). Its derivative is surprisingly simple but very important.

The Rule: If \( y = \ln x \), then \( \frac{dy}{dx} = \frac{1}{x} \).

Common Mistake Alert: Students often try to apply the Power Rule to \( \ln x \). Remember, \( \ln x \) is not \( x^n \). It has its own special rule!

Key Takeaway: The derivative of \( \ln x \) is always \( \frac{1}{x} \) (for \( x > 0 \)).

4. Trigonometric Functions: \( \sin, \cos, \) and \( \tan \)

When differentiating trig functions, there is one golden rule: Your calculator must be in Radians. Differentiation rules for trig functions do not work in degrees!

The Sine and Cosine Cycle

Sine and Cosine follow a predictable cycle when you differentiate them:

1. \( \sin(kx) \rightarrow k\cos(kx) \)
2. \( \cos(kx) \rightarrow -k\sin(kx) \)

Memory Aid: Think of a "Wheel of Differentiation":
\( \sin \rightarrow \cos \rightarrow -\sin \rightarrow -\cos \rightarrow \) (back to \( \sin \)).
Notice that "Cos" goes to "Minus" (C to M... like Cheese Melt!).

The Tangent Rule

Tangent is slightly different. Its derivative involves a function called secant (\( \sec \)), which is \( \frac{1}{\cos} \).

The Rule: If \( y = \tan(kx) \), then \( \frac{dy}{dx} = k\sec^2(kx) \).

Example: If \( y = \sin(4x) \), then \( \frac{dy}{dx} = 4\cos(4x) \).
Example: If \( y = \cos(2x) \), then \( \frac{dy}{dx} = -2\sin(2x) \).
Example: If \( y = \tan(3x) \), then \( \frac{dy}{dx} = 3\sec^2(3x) \).

Key Takeaway: Sine stays positive when turning into Cosine; Cosine turns negative when turning into Sine. Tangent becomes \( \sec^2 \).

5. Sums, Differences, and Constant Multiples

The syllabus requires you to differentiate combinations of these standard functions. Don't let a long equation scare you! You can simply differentiate each part one by one.

The Rules:
1. Constant Multiples: If there is a number in front of the function, it just stays there. (Derivative of \( 5x^2 \) is \( 5 \times 2x = 10x \)).
2. Sums/Differences: If functions are added or subtracted, differentiate them individually.

Step-by-Step Example:
Differentiate \( y = 4x^3 + 2e^{3x} - \sin(2x) \)
1. Differentiate \( 4x^3 \) using the power rule: \( 12x^2 \).
2. Differentiate \( 2e^{3x} \) (the 3 comes down): \( 2 \times 3e^{3x} = 6e^{3x} \).
3. Differentiate \( -\sin(2x) \) (the 2 comes down, sin becomes cos): \( -2\cos(2x) \).
Final Result: \( \frac{dy}{dx} = 12x^2 + 6e^{3x} - 2\cos(2x) \).

Quick Review: Common Derivatives Table

Function \( y \) | Derivative \( \frac{dy}{dx} \)
\( x^n \) | \( nx^{n-1} \)
\( e^{kx} \) | \( ke^{kx} \)
\( a^{kx} \) | \( k(a^{kx})\ln a \)
\( \ln x \) | \( \frac{1}{x} \)
\( \sin(kx) \) | \( k\cos(kx) \)
\( \cos(kx) \) | \( -k\sin(kx) \)
\( \tan(kx) \) | \( k\sec^2(kx) \)

Final Key Takeaway: Differentiation is a linear process. Break complex-looking expressions into their "standard" components, apply the specific rule for each, and then re-combine them. Consistency is key—practice these patterns until they feel like second nature!