#Power Rule and Differentiation

#Table of Contents

1. Introduction to the Power Rule
2. Differentiating Powers of x
   • Basic Differentiation
   • Examples
   • Fractional and Negative Powers
3. Worked Example
4. Derivatives of Sums, Differences, and Constant Multiples
   • Sums and Differences
   • Constant Multiples
   • Special Cases
5. Simplifying Expressions to Find Derivatives
   • Expansion
   • Using Laws of Exponents
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

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#1. Introduction to the Power Rule

The power rule is a fundamental tool in calculus for differentiating functions of the form $x^n$. It simplifies the process of finding the derivative of polynomials and other functions involving powers of $x$.

#2. Differentiating Powers of x

#Basic Differentiation

#Examples

1. Simple Power:

   • If $f(x) = x^7$, then: $$f'(x) = 7x^{6}$$

2. Fractional Power:

   • If $g(x) = x^{\frac{2}{3}}$, then: $$g'(x) = \frac{2}{3} x^{-\frac{1}{3}}$$

3. Negative Power:

   • If $h(x) = x^{-3}$, then: $$h'(x) = -3 x^{-4}$$

Using the power rule is much quicker than using the definition of a derivative, but you should still understand how to use the definition.

#3. Worked Example

Find the derivative of the function $f(x) = x^3$:

#(i) By Using the Power Rule

[ f(x) = x^3 \implies f'(x) = 3x^{2} ]

#(ii) By Using the Definition of a Derivative

[ f(x) = x^3 \implies f(x+h) = (x+h)^3 ]

[ f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - x^3}{h} ]

Expanding and simplifying:

[ f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} ]

[ f'(x) = \lim_{h \to 0} \left(3x^2 + 3xh + h^2\right) ]

As $h$ tends to zero, the terms containing $h$ will tend to zero:

[ f'(x) = 3x^2 ]

#4. Derivatives of Sums, Differences, and Constant Multiples

#Sums and Differences

When differentiating sums or differences of powers of $x$:

Example: If $f(x) = x^3 + x^7 - x^{12} + x^{\frac{1}{2}}$:

[ f'(x) = 3x^2 + 7x^6 - 12x^{11} + \frac{1}{2}x^{-\frac{1}{2}} ]

#Constant Multiples

Example: If $f(x) = 12x^4$:

[ f'(x) = 12 \times 4 x^{3} = 48 x^{3} ]

#Special Cases

1. Linear Functions:

   • If $f(x) = ax$, then $f'(x) = a$
     • Example: If $f(x) = 4x$, then $f'(x) = 4$

2. Constant Functions:

   • If $g(x) = a$, then $g'(x) = 0$
     • Example: If $g(x) = 2$, then $g'(x) = 0$

The function $f(x)$ is defined by $f(x) = 2x^{\frac{3}{2}} + 5x^{-3} - 9x + 7$. Find the derivative of $f(x)$.

Differentiate each term individually:

[ f'(x) = 2 \times \frac{3}{2} x^{\frac{1}{2}} + 5 \times -3 x^{-4} - 9 + 0 ]

Simplify:

[ f'(x) = 3 x^{\frac{1}{2}} - 15 x^{-4} - 9 ]

#5. Simplifying Expressions to Find Derivatives

#Expansion

If the function is not simply a sum of multiples of $\pm x^n$, it may need to be simplified before differentiation.

Example:

If $f(x) = (x + 2)(2x^2 + 5)$:

Expand the brackets:

[ f(x) = 2x^3 + 4x^2 + 5x + 10 ]

Differentiate each term:

[ f'(x) = 6x^2 + 8x + 5 ]

#Using Laws of Exponents

Example:

If $g(x) = \frac{9x^4 \cdot x^3}{x^2}$:

Simplify using laws of exponents:

[ g(x) = \frac{9x^7}{x^2} = 9x^5 ]

Differentiate:

[ g'(x) = 45x^4 ]

Differentiate $h(x) = \frac{2}{\sqrt{x}}$.

Rewrite using laws of exponents:

[ h(x) = 2x^{-\frac{1}{2}} ]

Differentiate:

[ h'(x) = -x^{-\frac{3}{2}} ]

#6. Practice Questions

Practice Question

Differentiate the following functions:

1. $f(x) = 3x^5 + 4x^{-2}$
2. $g(x) = \frac{7x^2 + 2}{x^3}$
3. $h(x) = (x - 1)(x^2 + 3)$
4. $k(x) = 5x^{\frac{3}{4}} - 8x^{-1} + 6$

#7. Glossary

• Derivative: The rate at which a function is changing at any given point.
• Power Rule: A rule for differentiating functions of the form $x^n$.
• Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.
• Sum Rule: The derivative of a sum of functions is the sum of the derivatives.
• Difference Rule: The derivative of a difference of functions is the difference of the derivatives.
• Exponent: The power to which a number or expression is raised.

#8. Summary and Key Takeaways

#Key Points

• The power rule simplifies differentiation of functions of the form $x^n$.
• Sum, difference, and constant multiple rules help in differentiating complex expressions.
• Simplification of expressions using expansion and laws of exponents is often necessary before differentiation.

#Key Takeaways

• Power Rule: $f'(x) = nx^{n-1}$
• Sum and Difference Rule: Differentiate each term separately.
• Constant Multiple Rule: Multiply the constant by the derivative of the function.
• Special Cases: Linear functions ($f(x) = ax$) and constant functions ($f(x) = a$).

Exam Tip

Always simplify the expression before differentiating to avoid mistakes.

#Exam Strategy

1. Identify the type of function: Determine if the function is a power of $x$, a sum, a difference, or a product.
2. Simplify if necessary: Expand or use laws of exponents.
3. Apply the power rule: Use the basic differentiation rules.
4. Check your work: Ensure that all steps follow logically and check for common mistakes.

Common Mistake

Avoid forgetting to apply the power rule correctly to fractional and negative exponents.

Exam Tip

This content directly links to the IB curriculum objectives of understanding and applying the concept and rules of differentiation.