Mathematics Master Notes: From Hardest to Easiest
Chapter 1 – Calculus: Differentiation
By Larry A
What is Differentiation?
Differentiation measures how one quantity changes in response to another. If y depends
on x, the derivative dy/dx tells us the rate of change of y with respect to x. Think of it as
how steep the graph is at a certain point.
Basic Rules
Rule Formula Example
Power Rule d(x■)/dx = n·x■■¹ d(x■)/dx = 4x³
Constant Rule d(c)/dx = 0 d(5)/dx = 0
Sum/Difference d(u±v)/dx = u'±v' d(x³+2x)/dx = 3x²+2
Product Rule d(uv)/dx = u'v + uv' d(x²sinx)/dx = 2xsinx + x²cosx
Quotient Rule d(u/v)/dx = (v·u'−u·v')/v² d(sinx/x)/dx = (xcosx−sinx)/x²
Chain Rule dy/dx = (dy/du)·(du/dx) d(sin(3x))/dx = 3cos(3x)
Special Functions
Function Derivative
sin x cos x
cos x -sin x
e■ e■
ln x 1/x
Applications
• Gradient of a Curve: slope = dy/dx
• Tangent & Normal Lines:
Tangent: y − y■ = m(x − x■)
Normal: gradient = −1/m
• Maxima and Minima:
1. Find dy/dx = 0
2. Use d²y/dx² to test: if >0 → minimum, if <0 → maximum
Chapter 1 – Calculus: Differentiation
By Larry A
What is Differentiation?
Differentiation measures how one quantity changes in response to another. If y depends
on x, the derivative dy/dx tells us the rate of change of y with respect to x. Think of it as
how steep the graph is at a certain point.
Basic Rules
Rule Formula Example
Power Rule d(x■)/dx = n·x■■¹ d(x■)/dx = 4x³
Constant Rule d(c)/dx = 0 d(5)/dx = 0
Sum/Difference d(u±v)/dx = u'±v' d(x³+2x)/dx = 3x²+2
Product Rule d(uv)/dx = u'v + uv' d(x²sinx)/dx = 2xsinx + x²cosx
Quotient Rule d(u/v)/dx = (v·u'−u·v')/v² d(sinx/x)/dx = (xcosx−sinx)/x²
Chain Rule dy/dx = (dy/du)·(du/dx) d(sin(3x))/dx = 3cos(3x)
Special Functions
Function Derivative
sin x cos x
cos x -sin x
e■ e■
ln x 1/x
Applications
• Gradient of a Curve: slope = dy/dx
• Tangent & Normal Lines:
Tangent: y − y■ = m(x − x■)
Normal: gradient = −1/m
• Maxima and Minima:
1. Find dy/dx = 0
2. Use d²y/dx² to test: if >0 → minimum, if <0 → maximum
