Unit II: Derivatives of Standard Functions and Rules
Derivatives of Standard Functions
Derivatives form the foundation of differential calculus, measuring the instantaneous rate of change of a function with respect to its independent variable. Below are the standard functions and their respective derivatives.
Polynomial Functions
A polynomial function generally takes the form f(x) = xn, where n is a real number. The derivative is found using the Power Rule.
Power Rule Formula:
If f(x) = xn, then df/dx = n × xn-1
Practical Example: Find the derivative of f(x) = x3.
Solution: Bring the exponent to the front and subtract 1 from the power: 3 × x3-1 = 3x2.
Real-World Application: In physics, if a position-time graph is represented by a polynomial equation, taking its derivative gives the velocity equation.
Exponential Functions
Exponential functions involve a constant base raised to a variable exponent, most notably the natural exponential function where the base is e.
Definition:
If f(x) = ex, then df/dx = ex
For a general exponential function f(x) = ax, df/dx = ax × ln(a)
Exam-Oriented Note: The derivative of ex is uniquely equal to itself, which is a frequently tested property in calculus problems.
Trigonometric Functions
Trigonometric functions relate the angles of a triangle to the lengths of its sides. Their derivatives are standard formulas that must be memorized for exams.
- If f(x) = sin(x), then df/dx = cos(x)
- If f(x) = cos(x), then df/dx = -sin(x)
- If f(x) = tan(x), then df/dx = sec2(x)
- If f(x) = csc(x), then df/dx = -csc(x) × cot(x)
- If f(x) = sec(x), then df/dx = sec(x) × tan(x)
- If f(x) = cot(x), then df/dx = -csc2(x)
Common Mistake: Forgetting the negative sign when differentiating co-functions (like cosine, cosecant, and cotangent). Always remember that derivatives of functions starting with 'c' yield negative results.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. The natural logarithm, ln(x), is standard in calculus.
Definition:
If f(x) = ln(x), then df/dx = 1 / x (for x > 0)
For a general logarithm f(x) = loga(x), df/dx = 1 / (x × ln(a))
Practical Example: Find the derivative of f(x) = ln(5x). Using properties of logarithms or chain rule concepts directly tied to standard expansions, the derivative simplifies to 1 / x.
Product and Quotient Rules for Derivatives
When functions are multiplied or divided rather than simply added or subtracted, specialized rules are required to find their derivatives.
Product Rule
The product rule is used when a function is formed by the multiplication of two differentiable functions, say u(x) and v(x).
Product Rule Formula:
If f(x) = u(x) × v(x), then
f'(x) = u(x) × v'(x) + v(x) × u'(x)
In words: "First times the derivative of the second, plus the second times the derivative of the first."
Practical Example: Find the derivative of f(x) = x2 × sin(x).
Step 1: Let u(x) = x2 and v(x) = sin(x).
Step 2: Find individual derivatives: u'(x) = 2x and v'(x) = cos(x).
Step 3: Apply the formula: x2 × cos(x) + sin(x) × 2x.
Quotient Rule
The quotient rule is applied when a function is expressed as a fraction or division of two differentiable functions.
Quotient Rule Formula:
If f(x) = u(x) / v(x), where v(x) ≠ 0, then
f'(x) = [v(x) × u'(x) - u(x) × v'(x)] / [v(x)]2
In words: "Low d-high minus high d-low, over the square of what's below."
Important Observation: Order matters heavily in the numerator of the quotient rule due to subtraction. Always start with the denominator function multiplied by the derivative of the numerator.
Comparison Summary of Derivative Rules
| Rule Name | Mathematical Structure | Derivative Formula |
|---|---|---|
| Product Rule | u(x) × v(x) | u × v' + v × u' |
| Quotient Rule | u(x) / v(x) | (v × u' - u × v') / v2 |
Concept of Higher Order Derivatives
The derivative of a function yields another function, which itself can often be differentiated. This process leads to the concept of higher order derivatives.
Definition:
- First Derivative: f'(x) or df/dx
- Second Derivative: f''(x) or d2f/dx2
- n-th Derivative: f(n)(x) or dnf/dxn
Step-by-Step Explanation:
1. Take the standard derivative of the original function to get the first derivative.
2. Take the derivative of the first derivative function to obtain the second derivative.
3. Continue this sequential differentiation process for higher orders as required.
Practical Example: Find the second derivative of f(x) = x3 + 5x2 - 4.
First Derivative: f'(x) = 3x2 + 10x
Second Derivative: f''(x) = 6x + 10
Real-World Application: In kinematics, if position is the zeroth derivative, velocity is the first derivative, and acceleration is the second derivative of position with respect to time.