Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. The derivative represents the rate of change or slope of a function at any given point. This comprehensive guide provides all essential differentiation formulas that students need for academic success.
Basic Differentiation Formulas
1. Fundamental Derivative Rules
| Rule Name |
Function f(x) |
Derivative f'(x) |
Explanation |
| Constant Rule |
c (constant) |
0 |
The derivative of any constant is zero |
| Power Rule |
xⁿ |
nxⁿ⁻¹ |
Multiply by the power, then reduce power by 1 |
| Constant Multiple |
cf(x) |
cf'(x) |
Constant can be factored out of derivative |
| Sum Rule |
f(x) + g(x) |
f'(x) + g'(x) |
Derivative of sum equals sum of derivatives |
| Difference Rule |
f(x) – g(x) |
f'(x) – g'(x) |
Derivative of difference equals difference of derivatives |
2. Product and Quotient Rules
| Rule Name |
Function |
Derivative |
Formula |
| Product Rule |
f(x)·g(x) |
f'(x)g(x) + f(x)g'(x) |
(uv)’ = u’v + uv’ |
| Quotient Rule |
f(x)/g(x) |
[f'(x)g(x) – f(x)g'(x)]/[g(x)]² |
(u/v)’ = (u’v – uv’)/v² |
3. Chain Rule
| Application |
Formula |
Expression |
| Chain Rule |
d/dx[f(g(x))] |
f'(g(x))·g'(x) |
| Composite Function |
y = f(u), u = g(x) |
dy/dx = (dy/du)·(du/dx) |
Derivatives of Elementary Functions
Algebraic Functions
| Function f(x) |
Derivative f'(x) |
Notes |
| x |
1 |
Linear function |
| x² |
2x |
Quadratic function |
| x³ |
3x² |
Cubic function |
| √x or x^(1/2) |
1/(2√x) |
Square root |
| 1/x or x⁻¹ |
-1/x² |
Reciprocal function |
| 1/x² or x⁻² |
-2/x³ |
Inverse square |
| ⁿ√x or x^(1/n) |
(1/n)x^((1/n)-1) |
nth root |
Trigonometric Functions
| Function f(x) |
Derivative f'(x) |
Domain Considerations |
| sin(x) |
cos(x) |
All real numbers |
| cos(x) |
-sin(x) |
All real numbers |
| tan(x) |
sec²(x) |
x ≠ (2n+1)π/2 |
| cot(x) |
-cosec²(x) |
x ≠ nπ |
| sec(x) |
sec(x)tan(x) |
x ≠ (2n+1)π/2 |
| cosec(x) |
-cosec(x)cot(x) |
x ≠ nπ |
Inverse Trigonometric Functions
| Function f(x) |
Derivative f'(x) |
Domain |
| sin⁻¹(x) or arcsin(x) |
1/√(1-x²) |
-1 < x < 1 |
| cos⁻¹(x) or arccos(x) |
-1/√(1-x²) |
-1 < x < 1 |
| tan⁻¹(x) or arctan(x) |
1/(1+x²) |
All real numbers |
| cot⁻¹(x) or arccot(x) |
-1/(1+x²) |
All real numbers |
| sec⁻¹(x) or arcsec(x) |
1/( |
x |
| cosec⁻¹(x) or arccosec(x) |
-1/( |
x |
Exponential Functions
| Function f(x) |
Derivative f'(x) |
Notes |
| eˣ |
eˣ |
Natural exponential |
| aˣ (a > 0) |
aˣ ln(a) |
General exponential |
| e^(kx) |
ke^(kx) |
k is constant |
Logarithmic Functions
| Function f(x) |
Derivative f'(x) |
Domain |
| ln(x) |
1/x |
x > 0 |
| log_a(x) |
1/(x ln(a)) |
x > 0, a > 0, a ≠ 1 |
| ln |
x |
| ln(f(x)) |
f'(x)/f(x) |
f(x) > 0 |
Hyperbolic Functions
| Function f(x) |
Derivative f'(x) |
Definition |
| sinh(x) |
cosh(x) |
(eˣ – e⁻ˣ)/2 |
| cosh(x) |
sinh(x) |
(eˣ + e⁻ˣ)/2 |
| tanh(x) |
sech²(x) |
sinh(x)/cosh(x) |
| coth(x) |
-cosech²(x) |
cosh(x)/sinh(x) |
| sech(x) |
-sech(x)tanh(x) |
1/cosh(x) |
| cosech(x) |
-cosech(x)coth(x) |
1/sinh(x) |
Advanced Differentiation Techniques
Implicit Differentiation
When y is defined implicitly in terms of x:
- Differentiate both sides with respect to x
- Treat y as a function of x (use chain rule)
- Solve for dy/dx
Example: For x² + y² = 25
- Differentiating: 2x + 2y(dy/dx) = 0
- Solving: dy/dx = -x/y
Logarithmic Differentiation
Used for complex products, quotients, or powers:
- Take natural logarithm of both sides
- Differentiate implicitly
- Solve for dy/dx
When to use: Functions of the form y = [f(x)]^(g(x))
Parametric Differentiation
For parametric equations x = f(t) and y = g(t):
| Derivative |
Formula |
| dy/dx |
(dy/dt)/(dx/dt) |
| d²y/dx² |
d/dt(dy/dx) ÷ (dx/dt) |
Higher Order Derivatives
| Notation |
Meaning |
Alternative Notation |
| f'(x) or dy/dx |
First derivative |
y’, D(y), d¹y/dx¹ |
| f”(x) or d²y/dx² |
Second derivative |
y”, D²(y) |
| f”'(x) or d³y/dx³ |
Third derivative |
y”’, D³(y) |
| f⁽ⁿ⁾(x) or dⁿy/dxⁿ |
nth derivative |
y⁽ⁿ⁾, Dⁿ(y) |
Special Differentiation Formulas
Derivative of Inverse Functions
If y = f(x) and x = f⁻¹(y), then: dx/dy = 1/(dy/dx) (provided dy/dx ≠ 0)
L’Hôpital’s Rule
For indeterminate forms (0/0 or ∞/∞): lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)
Common Differential Equations Formulas
| Type |
General Form |
Solution Method |
| Separable |
dy/dx = f(x)g(y) |
Separate variables and integrate |
| Linear First Order |
dy/dx + P(x)y = Q(x) |
Use integrating factor e^(∫P dx) |
| Homogeneous |
dy/dx = f(y/x) |
Substitute v = y/x |
| Exact |
M(x,y)dx + N(x,y)dy = 0 |
Check ∂M/∂y = ∂N/∂x |
Integration Formulas (Reverse of Differentiation)
Basic Integration Formulas
| Function |
Integral |
Notes |
| xⁿ |
xⁿ⁺¹/(n+1) + C |
n ≠ -1 |
| 1/x |
ln |
x |
| eˣ |
eˣ + C |
Natural exponential |
| aˣ |
aˣ/ln(a) + C |
a > 0, a ≠ 1 |
| sin(x) |
-cos(x) + C |
| cos(x) |
sin(x) + C |
| sec²(x) |
tan(x) + C |
| cosec²(x) |
-cot(x) + C |
| sec(x)tan(x) |
sec(x) + C |
| cosec(x)cot(x) |
-cosec(x) + C |
| 1/√(1-x²) |
sin⁻¹(x) + C |
| 1/(1+x²) |
tan⁻¹(x) + C |
| 1/(x√(x²-1)) |
sec⁻¹(x) + C |
Practical Applications of Differentiation
1. Rate of Change Problems
- Velocity: v = ds/dt (derivative of position)
- Acceleration: a = dv/dt = d²s/dt² (second derivative of position)
2. Optimization Problems
- Finding maximum and minimum values
- Critical points where f'(x) = 0
3. Curve Sketching
- Increasing/Decreasing: f'(x) > 0 (increasing), f'(x) < 0 (decreasing)
- Concavity: f”(x) > 0 (concave up), f”(x) < 0 (concave down)
- Inflection Points: Where f”(x) = 0 or undefined
Frequently Asked Questions about Differentiation Formulas
Q. What is the difference between differentiation and derivative?
Differentiation is the process of finding the derivative, while the derivative is the result—the function that represents the rate of change. Think of differentiation as the action and derivative as the outcome.
Q. What is the easiest way to remember differentiation formulas?
Practice is key, but these strategies help:
- Understand the logic behind each formula rather than memorizing blindly
- Use mnemonic devices (e.g., “cos goes negative” for d/dx[cos(x)] = -sin(x))
- Practice regularly with different function types
- Create formula sheets grouped by function families
Q. When do I use the product rule vs. the chain rule?
- Product Rule: When you have two functions multiplied together: f(x)·g(x)
- Chain Rule: When you have a composite function (function within a function): f(g(x))
- Sometimes you need both in the same problem
Q. What are the most common mistakes students make in differentiation?
- Forgetting to apply the chain rule for composite functions
- Incorrectly applying the power rule to exponential functions
- Sign errors in trigonometric derivatives (especially with cos and cot)
- Forgetting that the derivative of a constant is zero
- Mixing up product rule and quotient rule formulas
Q. How is differentiation used in real life?
- Physics: Calculating velocity, acceleration, and force
- Economics: Finding marginal cost, revenue, and profit
- Engineering: Optimizing designs and analyzing systems
- Medicine: Modeling disease spread and drug concentration
- Computer Science: Machine learning optimization algorithms
Q. What is the difference between dy/dx and d/dx?
- dy/dx is the derivative of y with respect to x (the result)
- d/dx is the differentiation operator (the action to be performed)
- Example: d/dx[x²] = 2x, so dy/dx = 2x when y = x²
Q. Can all functions be differentiated?
No. A function must be continuous and smooth at a point to be differentiable there. Functions are not differentiable at:
- Sharp corners or cusps
- Discontinuities or jumps
- Vertical tangent lines
Q. What is the relationship between differentiation and integration?
They are inverse operations according to the Fundamental Theorem of Calculus:
- Integration “undoes” differentiation
- Differentiation “undoes” integration
- If F'(x) = f(x), then ∫f(x)dx = F(x) + C
Q. How do I differentiate complex functions?
Break them down using:
- Identify the outermost operation
- Apply appropriate rules (chain, product, quotient)
- Work from outside to inside
- Simplify the final answer
Q. What is implicit differentiation and when is it used?
Implicit differentiation is used when y cannot be easily isolated. Common scenarios:
- Equations like x² + y² = r² (circles)
- Related rates problems
- Equations where solving for y is difficult or impossible
Q. What are higher-order derivatives used for?
- Second derivative (f”): Determines concavity and acceleration
- Third derivative (f”’): Jerk in physics (rate of change of acceleration)
- nth derivative: Pattern recognition, Taylor series, differential equations
Q. How do I know which differentiation technique to use?
Follow this decision tree:
- Is it a basic function? → Use standard formulas
- Is it a sum/difference? → Differentiate term by term
- Is it a product? → Product rule
- Is it a quotient? → Quotient rule
- Is it composite? → Chain rule
- Is y defined implicitly? → Implicit differentiation
- Is it very complex? → Consider logarithmic differentiation
