Introduction

In mathematics, differentiation is a fundamental concept that involves finding the rate at which a function changes. It is a crucial tool in calculus, allowing us to analyze functions and solve various real-world problems. The process of differentiation gives us the derivative of a function, which represents its instantaneous rate of change at any given point.

Definition of Differentiation

Differentiation is the process of finding the derivative of a function. The derivative of a function $f(x)$ with respect to $x$ is denoted by $f'(x)$ or $\frac{df}{dx}$. It represents the rate of change of $f(x)$ with respect to $x$.

Example:

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

Solution: To find the derivative, we differentiate each term of the function: $$ \begin{align*} f'(x) &= \frac{d}{dx}(3x^2) + \frac{d}{dx}(2x) - \frac{d}{dx}(1) \ &= 6x + 2 - 0 \ &= 6x + 2 \end{align*} $$

Rules of Differentiation

1. Constant Rule: The derivative of a constant is zero.

   If $f(x) = c$, where $c$ is a constant, then $f'(x) = 0$.

2. Power Rule: The derivative of $x^n$ is $nx^{n-1}$.

   If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

Example:

Find the derivative of the function $g(x) = 4x^3$.

Solution: Applying the power rule: $$ \begin{align*} g'(x) &= \frac{d}{dx}(4x^3) \ &= 4 \cdot 3x^{3-1} \ &= 12x^2 \end{align*} $$

3. Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives.

   If $f(x) = u(x) \pm v(x)$, then $f'(x) = u'(x) \pm v'(x)$.

4. Product Rule: The derivative of the product of two functions $u(x)$ and $v(x)$ is given by:

   $$(u \cdot v)' = u'v + uv'$$

Example:

Find the derivative of the function $h(x) = (2x^2 + 3x)(4x - 1)$.

Solution: Applying the product rule: $$ \begin{align*} h'(x) &= (2x^2 + 3x)'(4x - 1) + (2x^2 + 3x)(4x - 1)' \ &= (4x + 3)(4x - 1) + (2x^2 + 3x) \cdot 4 \ &= 16x^2 + 12x - 4 + 8x^2 + 12x \ &= 24x^2 + 24x - 4 \end{align*} $$

Common Mistakes

1. Forgetting to apply the rules of differentiation correctly.
2. Misinterpreting the derivative of a constant as zero without proper justification.
3. Incorrectly differentiating the product of functions using the product rule.

Key Points

• Differentiation is the process of finding the derivative of a function.
• Rules of differentiation include the constant rule, power rule, sum/difference rule, and product rule.
• Practice is essential to mastering the concept of differentiation.

Practice Questions

1. Find the derivative of the function $f(x) = 5x^4 - 2x^2 + 3$.

Solution: Applying the power rule: $$ \begin{align*} f'(x) &= \frac{d}{dx}(5x^4) - \frac{d}{dx}(2x^2) + \frac{d}{dx}(3) \ &= 5 \cdot 4x^{4-1} - 2 \cdot 2x^{2-1} + 0 \ &= 20x^3 - 4x \end{align*} $$

2. Determine the derivative of the function $h(x) = \frac{2}{x}$.

Solution: Applying the power rule and the constant rule: $$ \begin{align*} h'(x) &= \frac{d}{dx}\left(\frac{2}{x}\right) \ &= 2 \cdot \frac{d}{dx}(x^{-1}) \ &= 2 \cdot (-1)x^{-1-1} \ &= -2x^{-2} \end{align*} $$

3. Given $g(x) = e^x$, find its derivative.

Solution: The derivative of $e^x$ is simply $e^x$.

4. If $f(x) = (x^2 + 1)^3$, calculate $f'(x)$.

Solution: Applying the chain rule: $$ \begin{align*} f'(x) &= 3(x^2 + 1)^2 \cdot 2x \ &= 6x(x^2 + 1)^2 \end{align*} $$

5. Find the derivative of the function $y = \sin(x) \cos(x)$.

Solution: Using the product rule: $$ \begin{align*} \frac{dy}{dx} &= \cos(x) \cos(x) + (-\sin(x))\sin(x) \ &= \cos^2(x) - \sin^2(x) \end{align*} $$

6. Determine the derivative of $f(x) = \ln(x^2)$.

Solution: Applying the chain rule: $$ \begin{align*} f'(x) &= \frac{1}{x^2} \cdot 2x \ &= \frac{2}{x} \end{align*} $$