Differentiation

Introduction

In mathematics, differentiation is a fundamental concept that deals with finding the rate at which a function changes. It is a key tool in calculus and is used to determine slopes, rates of change, and concavity of functions. Differentiation allows us to analyze the behavior of functions and solve a variety of real-world problems. In this topic, we will explore the basic principles of differentiation and learn how to apply them to various functions.

Definition of Differentiation

Differentiation is the process of finding the derivative of a function. The derivative of a function $f(x)$, denoted by $f'(x)$ or $\frac{df}{dx}$, represents the rate of change of the function with respect to the independent variable $x$. Mathematically, the derivative of a function $f(x)$ is defined as: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Example 1: Find the derivative of $f(x) = 2x^2 + 3x - 4$.

To find the derivative of $f(x)$, we apply the definition of differentiation: $$f'(x) = \lim_{h \to 0} \frac{2(x+h)^2 + 3(x+h) - 4 - (2x^2 + 3x - 4)}{h}$$ Expanding and simplifying, we get: $$f'(x) = \lim_{h \to 0} \frac{4x + 2h + 3 - 4}{h}$$ $$f'(x) = \lim_{h \to 0} \frac{4x + 2h - 1}{h}$$ $$f'(x) = 4x - 1$$ Therefore, the derivative of $f(x) = 2x^2 + 3x - 4$ is $f'(x) = 4x - 1$.

Rules of Differentiation

There are several rules that govern the differentiation of functions. These rules enable us to find the derivatives of different types of functions efficiently. Some important rules include:

1. Power Rule: If $f(x) = x^n$, where $n$ is a constant, then $f'(x) = nx^{n-1}$.

Example 2: Find the derivative of $g(x) = 5x^3$.

Using the power rule, we have: $$g'(x) = 3 \cdot 5x^{3-1} = 15x^2$$ Therefore, the derivative of $g(x) = 5x^3$ is $g'(x) = 15x^2$.

2. Sum Rule: If $f(x) = u(x) + v(x)$, then $f'(x) = u'(x) + v'(x)$.

Example 3: Find the derivative of $h(x) = 3x^2 + 4x$.

Using the sum rule, we have: $$h'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(4x) = 6x + 4$$ Therefore, the derivative of $h(x) = 3x^2 + 4x$ is $h'(x) = 6x + 4$.

3. Product Rule: If $f(x) = u(x) \cdot v(x)$, then $f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$.

Example 4: Find the derivative of $k(x) = (2x + 1)(3x - 2)$.

Using the product rule, we have: $$k'(x) = (2)(3x - 2) + (2x + 1)(3) = 6x - 4 + 6x + 3 = 12x - 1$$ Therefore, the derivative of $k(x) = (2x + 1)(3x - 2)$ is $k'(x) = 12x - 1$.

4. Chain Rule: If $f(x) = g(u(x))$, then $f'(x) = g'(u(x)) \cdot u'(x)$.

Example 5: Find the derivative of $m(x) = (2x^2 + 1)^3$.

Let $u(x) = 2x^2 + 1$. Applying the chain rule, we have: $$m'(x) = 3(2x^2 + 1)^2 \cdot \frac{d}{dx}(2x^2 + 1)$$ $$m'(x) = 3(2x^2 + 1)^2 \cdot 4x = 12x(2x^2 + 1)^2$$ Therefore, the derivative of $m(x) = (2x^2 + 1)^3$ is $m'(x) = 12x(2x^2 + 1)^2$.

Common Mistakes

1. Forgetting to apply the rules of differentiation correctly, leading to incorrect derivative calculations.
2. Misinterpreting the chain rule and not properly identifying the inner and outer functions.
3. Not simplifying the final derivative expression, resulting in errors in the solution.

Key Points

• Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function.
• Important rules of differentiation include the power rule, sum rule, product rule, and chain rule.
• Practice is essential to mastering differentiation and applying it effectively in various mathematical problems.

Practice Questions

1. Find the derivative of $f(x) = 4x^3 - 2x^2 + 5x - 1$.
2. Determine the derivative of $g(x) = \frac{2}{x}$.
3. Calculate the derivative of $h(x) = e^{3x}$.
4. Given $f(x) = \sin(2x)$, find $f'(x)$.
5. If $g(x) = \log(x^2)$, what is $g'(x)$?

Practice Questions - Worked Solutions

1. Solution Given $f(x) = 4x^3 - 2x^2 + 5x - 1$, we find the derivative as follows: $$f'(x) = 3 \cdot 4x^{3-1} - 2 \cdot 2x^{2-1} + 5$$ $$f'(x) = 12x^2 - 4x + 5$$

2. Solution For $g(x) = \frac{2}{x}$, the derivative is calculated using the power rule: $$g'(x) = -2x^{-2} = -\frac{2}{x^2}$$

3. Solution To find the derivative of $h(x) = e^{3x}$, we apply the chain rule: $$h'(x) = 3e^{3x}$$

4. Solution Given $f(x) = \sin(2x)$, the derivative is: $$f'(x) = 2\cos(2x)$$

5. Solution For $g(x) = \log(x^2)$, the derivative is: $$g'(x) = \frac{2}{x}$$