Differentiation is solved by applying a set of rules and techniques to find the derivative of a function, which represents its instantaneous rate of change. Here's a breakdown of the process:
1. Understanding the Basics
- What is a derivative? The derivative of a function f(x), denoted as f'(x) or dy/dx, represents the slope of the tangent line to the function's graph at a specific point. It tells you how much the function's output changes for a tiny change in its input.
- Common Notation: You'll see these used interchangeably: f'(x), dy/dx, d/dx [f(x)].
2. Essential Differentiation Rules
These are the foundation for solving differentiation problems.
2.1 Power Rule
- Rule: If f(x) = xn, then f'(x) = nxn-1.
- Example: If f(x) = x3, then f'(x) = 3x2.
2.2 Constant Rule
- Rule: If f(x) = c (where c is a constant), then f'(x) = 0.
- Example: If f(x) = 5, then f'(x) = 0.
2.3 Constant Multiple Rule
- Rule: If f(x) = cu(x) (where c is a constant), then f'(x) = cu'(x).
- Example: If f(x) = 4x2, then f'(x) = 4(2x) = 8x.
2.4 Sum/Difference Rule
- Rule: If y = u(x) ± v(x), then dy/dx = du/dx ± dv/dx. In other words, the derivative of a sum or difference of functions is the sum or difference of their derivatives.
- Example: If f(x) = x3 + 2x, then f'(x) = 3x2 + 2.
2.5 Product Rule
- Rule: If y = u(x) × v(x), then dy/dx = u(x) dv/dx + v(x) du/dx.
- Example: If f(x) = x2sin(x), then f'(x) = x2cos(x) + sin(x)(2x).
2.6 Quotient Rule
- Rule: If y = u(x) ÷ v(x), then dy/dx = (v(x) du/dx - u(x) dv/dx) / (v(x))2.
- Example: If f(x) = sin(x) / x, then f'(x) = (xcos(x) - sin(x)1) / x2.
2.7 Chain Rule
- Rule: If y = f(g(x)), then dy/dx = f'(g(x)) g'(x)*. This is used for composite functions.
- Example: If f(x) = (x2 + 1)3, then f'(x) = 3(x2 + 1)2 (2x) = 6x(x2 + 1)2*.
3. Differentiation of Common Functions
| Function f(x) | Derivative f'(x) |
|---|---|
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec2(x) |
| ex | ex |
| ln(x) | 1/x |
4. Steps for Solving Differentiation Problems
- Identify the function: Clearly define the function f(x) you need to differentiate.
- Choose the appropriate rule(s): Determine which differentiation rule(s) apply based on the function's structure (power rule, product rule, quotient rule, chain rule, etc.).
- Apply the rule(s): Carefully apply the chosen rule(s), showing each step clearly.
- Simplify: Simplify the resulting expression to its simplest form.
5. Examples
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Example 1: Find the derivative of f(x) = 5x4 - 3x2 + 7.
- f'(x) = 5(4x3) - 3(2x) + 0 = 20x3 - 6x
-
Example 2: Find the derivative of f(x) = excos(x).
- Using the product rule: f'(x) = ex(-sin(x)) + cos(x)(ex) = ex(cos(x) - sin(x))
6. Advanced Techniques
- Implicit Differentiation: Used when y is not explicitly defined as a function of x.
- Logarithmic Differentiation: Useful for differentiating complex functions involving products, quotients, and exponents.
Conclusion
Solving differentiation problems involves understanding and applying a set of rules and techniques, starting with basic rules like the power rule and progressing to more complex rules like the product, quotient, and chain rules. Practice is key to mastering these techniques and becoming proficient in differentiation.
