🚀 Level 2 - Topic 2: Basic Differentiation Rules 🛠️

1) Introduction: Why We Need Differentiation Rules - Working Smarter, Not Harder

In the previous topic, we learned the fundamental limit definition of the derivative. While this definition is essential for understanding what a derivative *is*, using it directly to find derivatives every time can be time-consuming and algebraically intensive, especially for more complex functions.

Fortunately, mathematicians have developed a set of differentiation rules that allow us to find derivatives of many common functions much more efficiently, without having to go back to the limit definition each time. These rules are derived *from* the limit definition, but once established, we can apply them as shortcuts. This topic will equip you with these indispensable rules.

Mastering these basic rules is absolutely crucial. They are the foundation for all further differentiation techniques and applications in calculus. Think of them as your essential toolkit for calculus.

Analogy: From First Principles to Tools

Imagine building with wood. Initially, you might shape each piece directly with very basic tools. But soon, you'd develop saws, drills, planes, etc., to work much more efficiently and build more complex structures. Differentiation rules are like these specialized tools for calculus, making the process of finding derivatives systematic and manageable.

2) The Constant Rule: Derivative of a Constant is Zero

The simplest differentiation rule is the Constant Rule. It states that the derivative of any constant function is always zero.

Rule 2.1: Constant Rule

If \( f(x) = c \), where \( c \) is any constant number, then the derivative is:

\( \frac{d}{dx}[c] = 0 \) or \( f'(x) = 0 \)

In words: The rate of change of a constant function is always zero because a constant function's value never changes, no matter how \( x \) changes. Graphically, a constant function is a horizontal line, and horizontal lines have a slope of zero everywhere.

Examples of the Constant Rule

If \( f(x) = 5 \), then \( f'(x) = \frac{d}{dx}[5] = 0 \).
If \( y = -3 \), then \( \frac{dy}{dx} = \frac{d}{dx}[-3] = 0 \).
If \( g(x) = \sqrt{7} \), then \( g'(x) = \frac{d}{dx}[\sqrt{7}] = 0 \). (Remember, \( \sqrt{7} \) is just a constant number).
If \( h(x) = \pi \), then \( h'(x) = \frac{d}{dx}[\pi] = 0 \). ( \( \pi \) is also a constant).
If \( y = e^2 \), then \( \frac{dy}{dx} = \frac{d}{dx}[e^2] = 0 \). ( \( e^2 \) is approximately 7.389, a constant value).

3) The Power Rule: Differentiating Powers of \( x \)

The Power Rule is one of the most frequently used differentiation rules. It tells us how to differentiate functions of the form \( x^n \), where \( n \) is a constant exponent.

Rule 2.2: Power Rule

If \( f(x) = x^n \), where \( n \) is any real number, then the derivative is:

\( \frac{d}{dx}[x^n] = n x^{n-1} \) or \( f'(x) = n x^{n-1} \)

In words: To differentiate \( x^n \), bring the exponent \( n \) down to the front as a multiplier, and then reduce the exponent by 1. This rule works for any real number exponent \( n \) (positive integers, negative integers, fractions, etc.).

Examples of the Power Rule

\( f(x) = x^3 \): Using the Power Rule with \( n = 3 \), \( f'(x) = 3x^{3-1} = 3x^2 \). So, \( \frac{d}{dx}[x^3] = 3x^2 \).
\( y = x^7 \): With \( n = 7 \), \( \frac{dy}{dx} = 7x^{7-1} = 7x^6 \). So, \( \frac{d}{dx}[x^7] = 7x^6 \).
\( g(x) = x^{-2} \): With \( n = -2 \), \( g'(x) = -2x^{-2-1} = -2x^{-3} = -\frac{2}{x^3} \). So, \( \frac{d}{dx}[x^{-2}] = -\frac{2}{x^3} \).
\( y = \sqrt{x} \): First rewrite \( \sqrt{x} \) as \( x^{1/2} \). With \( n = \frac{1}{2} \), \( \frac{dy}{dx} = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}} \). So, \( \frac{d}{dx}[\sqrt{x}] = \frac{1}{2\sqrt{x}} \).
\( h(x) = \frac{1}{x^4} \): Rewrite \( \frac{1}{x^4} \) as \( x^{-4} \). With \( n = -4 \), \( h'(x) = -4x^{-4-1} = -4x^{-5} = -\frac{4}{x^5} \). So, \( \frac{d}{dx}\left[\frac{1}{x^4}\right] = -\frac{4}{x^5} \).

4) The Constant Multiple Rule: Dealing with Constant Factors

The Constant Multiple Rule helps us differentiate a function that is multiplied by a constant. It states that we can "pull out" the constant factor before differentiating.

Rule 2.3: Constant Multiple Rule

If \( c \) is a constant and \( f(x) \) is a differentiable function, then:

\( \frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)] = c \cdot f'(x) \) or \( (c \cdot f(x))' = c \cdot f'(x) \)

In words: When differentiating a constant times a function, you can keep the constant as a multiplier and just differentiate the function part.

Examples of the Constant Multiple Rule

\( y = 5x^2 \): Using the Constant Multiple Rule with \( c = 5 \) and \( f(x) = x^2 \), we get \( \frac{dy}{dx} = 5 \cdot \frac{d}{dx}[x^2] = 5 \cdot (2x) = 10x \). So, \( \frac{d}{dx}[5x^2] = 10x \).
\( g(x) = -3x^4 \): With \( c = -3 \) and \( f(x) = x^4 \), \( g'(x) = -3 \cdot \frac{d}{dx}[x^4] = -3 \cdot (4x^3) = -12x^3 \). So, \( \frac{d}{dx}[-3x^4] = -12x^3 \).
\( h(x) = \frac{x^3}{2} \): We can rewrite \( \frac{x^3}{2} \) as \( \frac{1}{2}x^3 \). Here \( c = \frac{1}{2} \) and \( f(x) = x^3 \). Then \( h'(x) = \frac{1}{2} \cdot \frac{d}{dx}[x^3] = \frac{1}{2} \cdot (3x^2) = \frac{3}{2}x^2 \). So, \( \frac{d}{dx}\left[\frac{x^3}{2}\right] = \frac{3}{2}x^2 \).
\( y = -\frac{4}{3}x^{-1} \): Here \( c = -\frac{4}{3} \) and \( f(x) = x^{-1} \). \( \frac{dy}{dx} = -\frac{4}{3} \cdot \frac{d}{dx}[x^{-1}] = -\frac{4}{3} \cdot (-1x^{-2}) = \frac{4}{3}x^{-2} = \frac{4}{3x^2} \). So, \( \frac{d}{dx}\left[-\frac{4}{3}x^{-1}\right] = \frac{4}{3x^2} \).

5) The Sum and Difference Rules: Differentiating Sums and Differences of Functions

The Sum Rule and Difference Rule are straightforward but incredibly useful for differentiating expressions that are sums or differences of multiple terms.

Rule 2.4: Sum Rule

If \( f(x) \) and \( g(x) \) are differentiable functions, then the derivative of their sum is the sum of their derivatives:

\( \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)] = f'(x) + g'(x) \) or \( (f(x) + g(x))' = f'(x) + g'(x) \)

Rule 2.5: Difference Rule

Similarly, the derivative of the difference of two differentiable functions is the difference of their derivatives:

\( \frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}[f(x)] - \frac{d}{dx}[g(x)] = f'(x) - g'(x) \) or \( (f(x) - g(x))' = f'(x) - g'(x) \)

In words: To differentiate a sum (or difference) of functions, differentiate each function term by term and then add (or subtract) the results. These rules extend to sums and differences of any number of functions.

Examples of the Sum and Difference Rules

\( y = x^3 + x^2 \): Using the Sum Rule, \( \frac{dy}{dx} = \frac{d}{dx}[x^3] + \frac{d}{dx}[x^2] = 3x^2 + 2x \). So, \( \frac{d}{dx}[x^3 + x^2] = 3x^2 + 2x \).
\( f(x) = x^4 - x \): Using the Difference Rule, \( f'(x) = \frac{d}{dx}[x^4] - \frac{d}{dx}[x] = 4x^3 - 1 \). (Remember \( \frac{d}{dx}[x] = \frac{d}{dx}[x^1] = 1 \cdot x^{1-1} = x^0 = 1 \)). So, \( \frac{d}{dx}[x^4 - x] = 4x^3 - 1 \).
\( g(x) = 2x^5 + 7x^3 - 4x + 9 \): Combining Sum, Difference, and Constant Multiple Rules: \( g'(x) = \frac{d}{dx}[2x^5] + \frac{d}{dx}[7x^3] - \frac{d}{dx}[4x] + \frac{d}{dx}[9] \) \( = 2\frac{d}{dx}[x^5] + 7\frac{d}{dx}[x^3] - 4\frac{d}{dx}[x] + \frac{d}{dx}[9] \) \( = 2(5x^4) + 7(3x^2) - 4(1) + 0 = 10x^4 + 21x^2 - 4 \). So, \( \frac{d}{dx}[2x^5 + 7x^3 - 4x + 9] = 10x^4 + 21x^2 - 4 \).
\( y = \sqrt{x} - \frac{3}{x^2} + 6 \): Rewrite using exponents: \( y = x^{1/2} - 3x^{-2} + 6 \). \( \frac{dy}{dx} = \frac{d}{dx}[x^{1/2}] - \frac{d}{dx}[3x^{-2}] + \frac{d}{dx}[6] \) \( = \frac{1}{2}x^{-1/2} - 3(-2x^{-3}) + 0 = \frac{1}{2\sqrt{x}} + 6x^{-3} = \frac{1}{2\sqrt{x}} + \frac{6}{x^3} \). So, \( \frac{d}{dx}\left[\sqrt{x} - \frac{3}{x^2} + 6\right] = \frac{1}{2\sqrt{x}} + \frac{6}{x^3} \).

6) Combining the Basic Rules: Differentiating Polynomials and Simple Expressions

With the Constant Rule, Power Rule, Constant Multiple Rule, Sum Rule, and Difference Rule, we can now efficiently differentiate any polynomial and many algebraic expressions. Polynomials are sums of terms of the form \( cx^n \), where \( c \) is a constant and \( n \) is a non-negative integer.

Example 6: Differentiating a Polynomial

Find the derivative of the polynomial \( P(x) = 4x^6 - 3x^4 + 2x^3 + 5x^2 - 7x + 8 \).

We apply the rules term by term: \( P'(x) = \frac{d}{dx}[4x^6] - \frac{d}{dx}[3x^4] + \frac{d}{dx}[2x^3] + \frac{d}{dx}[5x^2] - \frac{d}{dx}[7x] + \frac{d}{dx}[8] \) \( = 4\frac{d}{dx}[x^6] - 3\frac{d}{dx}[x^4] + 2\frac{d}{dx}[x^3] + 5\frac{d}{dx}[x^2] - 7\frac{d}{dx}[x] + \frac{d}{dx}[8] \) \( = 4(6x^5) - 3(4x^3) + 2(3x^2) + 5(2x) - 7(1) + 0 \) \( = 24x^5 - 12x^3 + 6x^2 + 10x - 7 \)

Therefore, \( P'(x) = 24x^5 - 12x^3 + 6x^2 + 10x - 7 \).

Example 7: Differentiating an Expression with Radicals and Rational Form

Find the derivative of \( y = \frac{2\sqrt{x}}{3} - \frac{5}{x^3} + 2\pi \).

Rewrite using exponents: \( y = \frac{2}{3}x^{1/2} - 5x^{-3} + 2\pi \).

Differentiate term by term: \( \frac{dy}{dx} = \frac{d}{dx}\left[\frac{2}{3}x^{1/2}\right] - \frac{d}{dx}[5x^{-3}] + \frac{d}{dx}[2\pi] \) \( = \frac{2}{3}\frac{d}{dx}[x^{1/2}] - 5\frac{d}{dx}[x^{-3}] + \frac{d}{dx}[2\pi] \) \( = \frac{2}{3}\left(\frac{1}{2}x^{-1/2}\right) - 5(-3x^{-4}) + 0 \) \( = \frac{1}{3}x^{-1/2} + 15x^{-4} = \frac{1}{3\sqrt{x}} + \frac{15}{x^4} \)

So, \( \frac{dy}{dx} = \frac{1}{3\sqrt{x}} + \frac{15}{x^4} \).

7) Practice Questions - Basic Differentiation Rules

Practice applying the Constant Rule, Power Rule, Constant Multiple Rule, Sum Rule, and Difference Rule.

Fundamental Practice Questions

Instructions: Find the derivative \( f'(x) \) or \( \frac{dy}{dx} \) for each function using the basic differentiation rules.

Q1. \( f(x) = x^6 \)
Q2. \( y = 12 \)
Q3. \( g(x) = 4x^3 \)
Q4. \( h(x) = x^7 - x^2 \)
Q5. \( y = 5x^4 + 3x^2 - 2x + 1 \)
Q6. \( f(x) = -x^{-3} \)
Q7. \( y = \frac{x^5}{7} \)
Q8. \( g(x) = 3\sqrt{x} \) (Rewrite \( \sqrt{x} \) as \( x^{1/2} \))
Q9. \( h(x) = \frac{4}{x^2} \) (Rewrite as \( 4x^{-2} \))
Q10. \( y = 2x^{3/2} - 5x^{-1/2} + \pi \)
Q11. \( f(x) = (x + 1)^2 \) (Expand first, then differentiate)
Q12. \( y = (x - 2)(x + 2) \) (Expand first)

Challenging Practice Questions

Instructions: These problems may require combining rules, algebraic simplification, or thinking conceptually.

Q1. Find the derivative of \( f(x) = \frac{x^4 - 3x^2 + 2\sqrt{x}}{x} \) (Hint: Simplify by dividing each term by \( x \) first).
Q2. For what value(s) of \( x \) does the graph of \( y = x^3 - 3x^2 + 2x - 1 \) have a horizontal tangent line? (Hint: Horizontal tangent means slope is zero, i.e., \( f'(x) = 0 \)).
Q3. If the position of a particle at time \( t \) is given by \( s(t) = 2t^3 - 9t^2 + 12t \), find the velocity \( v(t) = s'(t) \) and acceleration \( a(t) = v'(t) = s''(t) \) as functions of time \( t \).
Q4. Find the equation of the tangent line to the curve \( y = 3x^2 - 5x + 2 \) at the point \( x = 1 \). (You'll need the point on the curve and the slope from the derivative).
Q5. A function is given by \( F(x) = ax^2 + bx + c \). Given that \( F'(1) = 5 \) and \( F'(-1) = -1 \), and \( F(0) = 3 \), find the values of the constants \( a \), \( b \), and \( c \).

8) Summary & Cheat Sheet - Basic Differentiation Rules

Let's summarize the essential basic differentiation rules for quick reference.

8.1) Basic Differentiation Rules - Cheat Sheet

Here are the core rules we've covered:

Constant Rule: \( \frac{d}{dx}[c] = 0 \) (where \( c \) is constant)
Power Rule: \( \frac{d}{dx}[x^n] = n x^{n-1} \) (for any real number \( n \))
Constant Multiple Rule: \( \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) \) (where \( c \) is constant)
Sum Rule: \( \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \)
Difference Rule: \( \frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x) \)

8.2) How to Apply Basic Rules - Strategy

Identify the Structure: Recognize if your function is a sum, difference, constant multiple, or power of \( x \) (or a combination).
Apply Rules Term-by-Term: Use Sum and Difference Rules to break down complex expressions into simpler terms.
Use Constant Multiple Rule: Pull out constant factors before differentiating variable parts.
Apply Power Rule: For terms of the form \( x^n \), use the Power Rule. Remember it works for negative and fractional exponents too.
Constant Rule for Constants: The derivative of any term that is just a constant is zero.
Simplify: After differentiating, simplify the resulting expression algebraically if needed.