Differentiation of a Basic Function

Differentiation of a Basic Function

Differentiation of Basic Functions

Definition and Rules

• Derivatives represent the rate of change of a function. They are a fundamental tool in calculus.
• The derivative of a constant, for example, c, is 0. d/dx(c) = 0
• The derivative of x with respect to x is 1. d/dx(x) = 1
• The power rule: The derivative of x^n, with respect to x, is n*x^(n-1).
• The derivative of e^x with respect to x is e^x.
• The derivative of ln(x), with natural log base e, with respect to x is 1/x.

Chain Rule

• The chain rule is used when you have a function within a function, also known as a composite function.
• If u = f(x) and v = g(x), then d/dx[f(g(x))] = f’(g(x))*g’(x).

Sum and Difference Rule

• The derivative of the sum (or difference) of two functions is the sum (or difference) of their derivatives.
• If u = f(x) and v = g(x), then d/dx(u + v) = du/dx + dv/dx.

Product Rule

• The derivative of the product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first.
• If u = f(x) and v = g(x), then d/dx(uv)= u(dv/dx) + v(du/dx).

Quotient Rule

• The derivative of the quotient of two functions is the bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared.
• If u = f(x) and v = g(x), then d/dx(u/v)= (v(du/dx) - u(dv/dx))/v^2.