Differential calculus is the branch of mathematics that deals with the rate of change of a function. The derivative describes how a function changes with respect to its input variable.
If y = f(x), the derivative is written as:
dy/dx or f′(x)
Basic Definition of Derivative
f′(x) = lim (h → 0) [f(x + h) − f(x)] / h
Standard Derivatives
- d/dx (c) = 0
- d/dx (x) = 1
- d/dx (xⁿ) = n xⁿ⁻¹
- d/dx (eˣ) = eˣ
- d/dx (aˣ) = aˣ ln(a)
- d/dx (log x) = 1/x
- d/dx (ln x) = 1/x
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec²x
- d/dx (cot x) = −csc²x
- d/dx (sec x) = sec x tan x
- d/dx (csc x) = −csc x cot x
Rules of Differentiation
1. Constant Multiple Rule
If y = k·f(x), then:
dy/dx = k f′(x)
2. Sum Rule
d/dx [f(x) + g(x)] = f′(x) + g′(x)
3. Difference Rule
d/dx [f(x) − g(x)] = f′(x) − g′(x)
Product Rule
If y = u·v, then:
dy/dx = u dv/dx + v du/dx
Quotient Rule
If y = u/v, then:
dy/dx = (v du/dx − u dv/dx) / v²
Chain Rule (Derivative of Composite Functions)
If y = f(g(x)), then:
dy/dx = f′(g(x)) · g′(x)
Parametric Differentiation
When x and y are given in terms of a parameter t:
dy/dx = (dy/dt) / (dx/dt)
Implicit Differentiation
Used when y cannot be easily isolated. Differentiate both sides treating y as a function of x, and multiply y-terms by dy/dx.
Logarithmic Differentiation
Useful for products, quotients and functions of the form xˣ or complicated expressions.
Steps:
- Take natural log on both sides.
- Use log rules to simplify.
- Differentiate using chain rule.
- Solve for dy/dx.
Successive Differentiation
Higher-order derivatives are:
- First derivative: f′(x)
- Second derivative: f″(x)
- Third derivative: f‴(x)
- And so on.
These are used to study curvature, acceleration, maxima, minima, and concavity.
Derivatives of Exponential and Logarithmic Functions
- d/dx (eˣ) = eˣ
- d/dx (eᵏˣ) = k eᵏˣ
- d/dx (aˣ) = aˣ ln(a)
- d/dx (logₐ x) = 1/(x ln a)
Derivatives of Trigonometric Functions
- sin x → cos x
- cos x → −sin x
- tan x → sec²x
- cot x → −csc²x
- sec x → sec x tan x
- csc x → −csc x cot x
Derivatives of Inverse Trigonometric Functions
- d/dx (sin⁻¹x) = 1/√(1 − x²)
- d/dx (cos⁻¹x) = −1/√(1 − x²)
- d/dx (tan⁻¹x) = 1/(1 + x²)
- d/dx (cot⁻¹x) = −1/(1 + x²)
Examples (Conceptual Overview)
- Using product rule to differentiate algebraic expressions
- Chain rule for composite functions like sin(3x² + 1)
- Quotient rule for rational functions
- Parametric derivatives using t
- Logarithmic differentiation for complicated products
Important Derivative Shortcuts
- (xⁿ)’ = nxⁿ⁻¹
- (aˣ)’ = aˣ ln a
- (ln x)’ = 1/x
- (eˣ)’ = eˣ
- (xᵃ)’ = a xᵃ⁻¹
Practice Questions
- Differentiate: y = (x² + 1)(x³ − 2)
- Find dy/dx if y = (3x + 4)/(2x² − 5)
- Differentiate y = sin(x²)
- Find dy/dx if x = t² + 1 and y = t³
- Use logarithmic differentiation for y = xˣ
Simple and structured notes for easy learning of differential calculus.
Detailed Notes:
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