Rules for Differentiation

Rules for Differentiation

• Determining the derivative of a function from first principles requires a long calculation and is prone to mistakes.
• The general rule for differentiation is: d/dx [x^n] = nx^(n-1), where n is a real number and n is not equal to 0.

Derivative Rules

• Derivative of a constant: d/dx [k] = 0.
• Derivative of a constant multiplied by a function: d/dx [k * f(x)] = k * d/dx [f(x)].
• Derivative of a sum: d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)].
• Derivative of a difference: d/dx [f(x) - g(x)] = d/dx [f(x)] - d/dx [g(x)].

When to Use the Rules for Differentiation

• Use the rules for differentiation when the question does not specify how to determine the derivative.
• Use first principles if specifically requested or if asked to differentiate using the definition of a derivative.

Notation

• Alternative notations for the derivative include: f'(x), y', dy/dx, df/dx, d/dx [f(x)], Df(x), and D_xy.
• D and d/dx are called differential operators because they indicate the operation of differentiation.
• dy/dx means y differentiated with respect to x.