Differentiation Calculator
Find derivatives with step-by-step solutions
Quick Select
Common Differentiation Formulas
d/dx(x^n) = n·x^(n-1)
Power Rule
d/dx(e^x) = e^x
Exponential
d/dx(sin(x)) = cos(x)
Trigonometric
d/dx(ln(x)) = 1/x
Logarithmic
What is Differentiation?
Differentiation finds the rate of change of a function. It calculates the slope of tangent lines and instantaneous velocity.
Key Rules
Power: d/dx(x^n) = n·x^(n-1)
Product: d/dx(fg) = f'g + fg'
Quotient: d/dx(f/g) = (f'g - fg')/g²
Chain: d/dx(f(g(x))) = f'(g(x))·g'(x)
Applications
- Physics: velocity (derivative of position), acceleration (derivative of velocity)
- Economics: marginal cost and revenue
- Engineering: optimization problems
- Biology: rate of population growth
Complete Guide to Differentiation
1. Definition of Differentiation
Differentiation is a fundamental concept in calculus that measures the instantaneous rate at which a quantity changes. Mathematically, if y = f(x) is a function, its derivative is defined as the limit:
This limit, when it exists, gives us the derivative of the function. The process of finding this derivative is called differentiation. The derivative represents how the output value of a function changes as its input changes, making it essential for understanding rates of change in various scientific and engineering contexts.
2. Geometric Meaning: Slope of Tangent
Geometrically, the derivative at a point represents the slope of the tangent line to the curve at that point. If you draw a line touching the curve at exactly one point, the slope of that line tells you the direction and steepness of the curve at that location. A positive slope indicates the function is increasing, a negative slope indicates it is decreasing, and a zero slope indicates a local maximum or minimum (turning point). This geometric interpretation is crucial for understanding curves, optimization problems, and motion analysis.
The derivative also measures concavity: where f''(x) > 0, the curve is concave upward (shaped like a cup); where f''(x) < 0, the curve is concave downward (shaped like a cap). Points where concavity changes are called inflection points.
3. All Differentiation Rules
3.1 Power Rule
For any real number n:
Examples: d/dx(x^5) = 5x^4, d/dx(x^(-3)) = -3x^(-4), d/dx(x^(1/2)) = (1/2)x^(-1/2)
3.2 Product Rule
For the product of two functions f(x) and g(x):
Memory aid: "First times derivative of second, plus second times derivative of first"
3.3 Quotient Rule
For the quotient of two functions f(x) and g(x):
Memory aid: "Low d-high minus high d-low, over low squared"
3.4 Chain Rule
For composite functions f(g(x)):
If y = (3x + 5)^4, then dy/dx = 4(3x + 5)^3 · 3 = 12(3x + 5)^3
4. Trigonometric Derivatives
d/dx(sin x) = cos x
d/dx(cos x) = -sin x
d/dx(tan x) = sec² x
d/dx(cot x) = -cosec² x
d/dx(sec x) = sec x tan x
d/dx(cosec x) = -cosec x cot x
5. Exponential and Logarithmic Derivatives
d/dx(e^x) = e^x
e is Euler's number (2.718...)
d/dx(a^x) = a^x · ln(a)
For any base a > 0
d/dx(ln x) = 1/x
Natural logarithm
d/dx(logₐ x) = 1/(x·ln(a))
Logarithm to any base
6. Applications
6.1 Velocity and Acceleration (Physics)
In physics, differentiation has profound applications:
- Position s(t) describes where an object is at time t
- Velocity v(t) = ds/dt is the rate of change of position (instantaneous speed with direction)
- Acceleration a(t) = dv/dt = d²s/dt² is the rate of change of velocity
If s(t) = 5t³ - 3t² + 2t, then v(t) = 15t² - 6t + 2 and a(t) = 30t - 6
6.2 Optimization Problems
Differentiation helps find maximum and minimum values of functions:
- Find critical points where f'(x) = 0 or f'(x) is undefined
- Use the second derivative test: f''(x) > 0 indicates local minimum, f''(x) < 0 indicates local maximum
- Check endpoints for absolute maximum/minimum in closed intervals
6.3 Economics Applications
Marginal cost = dC/dq (derivative of cost with respect to quantity), helping businesses determine optimal production levels. Marginal revenue similarly guides pricing decisions.
7. Solved Examples from NCERT
Example 1: Find dy/dx if y = x³ + 2x² - 5x + 3
Solution:
Using power rule for each term:
dy/dx = 3x² + 4x - 5 + 0 = 3x² + 4x - 5
Example 2: Differentiate y = (2x + 3)(x² - 1)
Solution:
Using product rule: f(x) = (2x + 3), g(x) = (x² - 1)
f'(x) = 2, g'(x) = 2x
dy/dx = 2(x² - 1) + (2x + 3)(2x) = 2x² - 2 + 4x² + 6x = 6x² + 6x - 2
Example 3: Find derivative of y = sin(3x² + 2)
Solution:
Using chain rule: outer f(u) = sin(u), inner u = 3x² + 2
f'(u) = cos(u), u' = 6x
dy/dx = cos(3x² + 2) · 6x = 6x cos(3x² + 2)
Example 4: Differentiate y = (x² + 1)/(x + 2)
Solution:
Using quotient rule: f = x² + 1, g = x + 2
f' = 2x, g' = 1
dy/dx = [(2x)(x + 2) - (x² + 1)(1)] / (x + 2)²
= [2x² + 4x - x² - 1] / (x + 2)² = (x² + 4x - 1) / (x + 2)²
Example 5: If a particle's position is s(t) = 4t³ - 3t² + 2t - 5, find velocity at t = 2 seconds
Solution:
v(t) = ds/dt = 12t² - 6t + 2
At t = 2: v(2) = 12(4) - 6(2) + 2 = 48 - 12 + 2 = 38 units/sec
8. Common Mistakes to Avoid
- Chain rule errors: Always differentiate the inner function. For (3x + 1)⁴, derivative is 4(3x + 1)³ × 3, not just 4(3x + 1)³
- Sign errors in quotient rule: Remember "low d-high minus high d-low" with the subtraction in the correct order
- Power rule with negative exponents: d/dx(x⁻¹) = -x⁻², not x⁻² (the negative stays negative)
- Confusing rules: Product rule has addition (+), quotient rule has subtraction (-)
- Forgetting the chain rule: tan(x²) is NOT sec²(x²); it is sec²(x²) × 2x
- Derivative of constant: Any constant (5, π, etc.) differentiates to 0
- Derivative of ln|x|: Remember the absolute value: d/dx(ln|x|) = 1/x for x ≠ 0
Practice Tips
- Always identify the structure of the function first: Is it a power, product, quotient, or composite?
- For complex functions, work step-by-step from the outside in
- Simplify your answer whenever possible
- Practice the chain rule extensively, as it appears in most real-world applications
- Memorize standard derivatives of trigonometric and exponential functions