Derivatives

What is a Derivative?

A derivative measures the rate at which a function changes as its input changes. In other words, it tells you how fast something is changing.

Notation:

The derivative of a function f(x) is denoted by f'(x) or dy/dx (if the function is y = f(x)).

Basic Rules:

1. Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)

* Example: f(x) = x^2, so f'(x) = 2x

1. Sum Rule: If f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x)

* Example: f(x) = 2x + 3, so f'(x) = 2 (since the derivative of a constant is 0)

1. Product Rule: If f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x)

* Example: f(x) = x^2 sin(x), so f'(x) = (2x)sin(x) + x^2 cos(x)

More Rules:

1. Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) \* g'(x)

* Example: y = sin(2x), so dy/dx = 2 cos(2x)

1. Quotient Rule: If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x)) / h(x)^2

* Example: f(x) = x^2 / sin(x), so f'(x) = ((sin(x))(2x) - x^2(cos(x))) / sin(x)^2

Important Notes:

1. The derivative of a constant is always 0.
2. The derivative of an exponential function (e.g., e^x, 3^x) is the same as the original function multiplied by its exponent.

I hope this summary and the examples help you understand derivatives better!