Enhanced Calculus Calculator

Essential Derivative Rules

Basic Rules

Power Rule

d/dx[x^n] = n·x^(n-1)

Constant Rule

d/dx[c] = 0

Constant Multiple

d/dx[c·f(x)] = c·f'(x)

Sum Rule

d/dx[f(x) + g(x)] = f'(x) + g'(x)

Advanced Rules

Product Rule

d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

Quotient Rule

d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)]/[g(x)]²

Chain Rule

d/dx[f(g(x))] = f'(g(x))·g'(x)

Implicit Differentiation

For F(x,y) = 0: dy/dx = -F_x/F_y

Common Derivatives

Exponential

d/dx[e^x] = e^x
d/dx[a^x] = a^x·ln(a)

Logarithmic

d/dx[ln(x)] = 1/x
d/dx[log_a(x)] = 1/(x·ln(a))

Trigonometric

d/dx[sin(x)] = cos(x)
d/dx[cos(x)] = -sin(x)
d/dx[tan(x)] = sec²(x)

Inverse Trig

d/dx[arcsin(x)] = 1/√(1-x²)
d/dx[arctan(x)] = 1/(1+x²)

Essential Integration Rules

Basic Integration Rules

Power Rule

∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)

Constant Rule

∫ c dx = cx + C

Sum Rule

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

Constant Multiple

∫ c·f(x) dx = c·∫ f(x) dx

Advanced Techniques

Substitution

∫ f(g(x))·g'(x) dx = ∫ f(u) du where u = g(x)

Integration by Parts

∫ u dv = uv - ∫ v du

Partial Fractions

For rational functions: decompose then integrate

Fundamental Theorem

∫[a to b] f(x) dx = F(b) - F(a)

Common Integrals

Exponential

∫ e^x dx = e^x + C
∫ a^x dx = a^x/ln(a) + C

Logarithmic

∫ 1/x dx = ln|x| + C
∫ ln(x) dx = x·ln(x) - x + C

Trigonometric

∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ sec²(x) dx = tan(x) + C

Inverse Trig

∫ 1/√(a²-x²) dx = arcsin(x/a) + C
∫ 1/(a²+x²) dx = (1/a)arctan(x/a) + C

Limit Evaluation Techniques

Basic Limit Laws

lim[x→a] [f(x) ± g(x)] = lim[x→a] f(x) ± lim[x→a] g(x)
lim[x→a] [f(x) · g(x)] = lim[x→a] f(x) · lim[x→a] g(x)
lim[x→a] [f(x) / g(x)] = lim[x→a] f(x) / lim[x→a] g(x) (if lim g(x) ≠ 0)

Indeterminate Forms

0/0: Use L'Hôpital's Rule, factoring, or rationalization
∞/∞: Use L'Hôpital's Rule or divide by highest power
0·∞: Rewrite as a fraction to get 0/0 or ∞/∞
∞-∞: Factor out common terms or rationalize
1^∞, 0^0, ∞^0: Use logarithms and exponentials

L'Hôpital's Rule

If lim[x→a] f(x)/g(x) = 0/0 or ∞/∞, then:
lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)

Important Limits

Trigonometric

lim[x→0] sin(x)/x = 1
lim[x→0] (1-cos(x))/x = 0

Exponential

lim[x→∞] (1+1/x)^x = e
lim[x→0] (e^x-1)/x = 1

Logarithmic

lim[x→0+] ln(x)/x = -∞
lim[x→0+] x·ln(x) = 0

Real-World Applications of Calculus

Physics & Engineering

Motion Analysis: Position, velocity, and acceleration relationships
Optimization: Minimizing cost, maximizing efficiency
Related Rates: How quantities change with respect to each other
Work and Energy: Calculating work done by variable forces
Electric Circuits: Current, voltage, and charge relationships

Economics & Business

Marginal Analysis: Marginal cost, revenue, and profit
Elasticity: Price elasticity of demand
Growth Models: Population and economic growth
Optimization: Maximum profit, minimum cost problems

Biology & Medicine

Population Dynamics: Modeling population growth and decay
Drug Concentration: Medicine absorption and elimination
Epidemic Models: Disease spread modeling
Biological Rhythms: Circadian cycles and periodic functions

Worked Examples

Example 1: Product Rule

Problem: Find the derivative of f(x) = x²·sin(x)

Solution:
• Using the product rule: (uv)' = u'v + uv'
• Let u = x² and v = sin(x)
• u' = 2x and v' = cos(x)
• f'(x) = (2x)·sin(x) + x²·cos(x)
• f'(x) = 2x·sin(x) + x²·cos(x)

Answer: f'(x) = 2x·sin(x) + x²·cos(x)

Example 2: Chain Rule

Problem: Find the derivative of f(x) = sin(3x² + 1)

Solution:
• Using the chain rule: [f(g(x))]' = f'(g(x))·g'(x)
• Outer function: f(u) = sin(u), f'(u) = cos(u)
• Inner function: g(x) = 3x² + 1, g'(x) = 6x
• f'(x) = cos(3x² + 1)·6x
• f'(x) = 6x·cos(3x² + 1)

Answer: f'(x) = 6x·cos(3x² + 1)

Example 3: Integration by Substitution

Problem: Evaluate ∫ 2x·√(x² + 1) dx

Solution:
• Let u = x² + 1, then du = 2x dx
• The integral becomes ∫ √u du
• ∫ u^(1/2) du = (2/3)u^(3/2) + C
• Substitute back: (2/3)(x² + 1)^(3/2) + C

Answer: (2/3)(x² + 1)^(3/2) + C

Example 4: L'Hôpital's Rule

Problem: Evaluate lim[x→0] (sin(x) - x)/x³

Solution:
• Direct substitution gives 0/0 (indeterminate)
• Apply L'Hôpital's rule: differentiate numerator and denominator
• lim[x→0] (cos(x) - 1)/(3x²) = 0/0 (still indeterminate)
• Apply L'Hôpital's rule again: lim[x→0] (-sin(x))/(6x) = 0/0
• Apply once more: lim[x→0] (-cos(x))/6 = -1/6

Answer: -1/6

Derivatives

Integrals

Limits

Series & Sequences

Calculus Reference & Learning

Essential Derivative Rules

Basic Rules

Advanced Rules

Common Derivatives

Essential Integration Rules

Basic Integration Rules

Advanced Techniques

Common Integrals

Limit Evaluation Techniques

Basic Limit Laws

Indeterminate Forms

L'Hôpital's Rule

Important Limits

Real-World Applications of Calculus

Physics & Engineering

Economics & Business

Biology & Medicine

Worked Examples

Example 1: Product Rule

Example 2: Chain Rule

Example 3: Integration by Substitution

Example 4: L'Hôpital's Rule