Calculus - Differentiation Basics Notes
1. What is Calculus?
Calculus is the branch of mathematics that studies how things change. It focuses on rates of change
(differential calculus) and accumulation of quantities (integral calculus).
2. What is Differentiation?
Differentiation is the process of finding the derivative of a function. It measures the rate at which a quantity
changes with respect to another.
3. Basic Rules of Differentiation:
Let f(x) and g(x) be differentiable functions and c be a constant:
- Constant Rule: d(c)/dx = 0
- Power Rule: d(x^n)/dx = n * x^(n-1)
- Sum Rule: d(f + g)/dx = df/dx + dg/dx
- Product Rule: d(fg)/dx = f * dg/dx + g * df/dx
- Quotient Rule: d(f/g)/dx = (g * df/dx - f * dg/dx) / g^2
- Chain Rule: d(f(g(x)))/dx = f'(g(x)) * g'(x)
4. Derivatives of Common Functions:
- d(sin x)/dx = cos x
- d(cos x)/dx = -sin x
- d(tan x)/dx = sec^2 x
- d(e^x)/dx = e^x
- d(ln x)/dx = 1/x
5. Applications of Derivatives:
- Finding slope of a curve at a point
- Determining increasing/decreasing nature of functions
- Finding maxima and minima
- Solving problems of motion, growth and decay
- Optimization problems in real life
1. What is Calculus?
Calculus is the branch of mathematics that studies how things change. It focuses on rates of change
(differential calculus) and accumulation of quantities (integral calculus).
2. What is Differentiation?
Differentiation is the process of finding the derivative of a function. It measures the rate at which a quantity
changes with respect to another.
3. Basic Rules of Differentiation:
Let f(x) and g(x) be differentiable functions and c be a constant:
- Constant Rule: d(c)/dx = 0
- Power Rule: d(x^n)/dx = n * x^(n-1)
- Sum Rule: d(f + g)/dx = df/dx + dg/dx
- Product Rule: d(fg)/dx = f * dg/dx + g * df/dx
- Quotient Rule: d(f/g)/dx = (g * df/dx - f * dg/dx) / g^2
- Chain Rule: d(f(g(x)))/dx = f'(g(x)) * g'(x)
4. Derivatives of Common Functions:
- d(sin x)/dx = cos x
- d(cos x)/dx = -sin x
- d(tan x)/dx = sec^2 x
- d(e^x)/dx = e^x
- d(ln x)/dx = 1/x
5. Applications of Derivatives:
- Finding slope of a curve at a point
- Determining increasing/decreasing nature of functions
- Finding maxima and minima
- Solving problems of motion, growth and decay
- Optimization problems in real life
