Sum and difference rules

Learning Objectives State the sum rule for derivatives in formal notation. State the difference rule for derivatives in formal notation. Apply the sum rule to find the derivative of a function composed of the sum of two or more terms. Apply the difference rule to find the derivative of a function composed of the difference of two terms. Combine the sum, difference, power, and constant multiple rules to differentiate any polynomial function. Algebraically manipulate functions with radicals or rational expressions into a sum or difference of power functions before differentiating. How can we find the rate of change of a company's total profit if we know the rates for its revenue and its costs? 📈 This tutorial introduces the Sum and Difference Rules, two of the most fund...

TermDefinitionExample DerivativeThe instantaneous rate of change of a function with respect to one of its variables. Geometrically, it represents the slope of the tangent line to the function's graph at a specific point.For the function f(x) = x², the derivative is f'(x) = 2x. Differentiable FunctionA function for which the derivative exists at each point in its domain. Smooth, continuous functions without sharp corners or vertical tangents are generally differentiable.Any polynomial function, like f(x) = 4x³ - 2x + 5, is differentiable everywhere. Power RuleA shortcut for finding the derivative of a variable raised to a power. The rule is d/dx[xⁿ] = nxⁿ⁻¹.The derivative of x⁵ is 5x⁴. Constant Multiple RuleThe derivative of a constant multiplied by a function is the constant mul...

The Sum Rule \frac{d}{dx} [f(x) + g(x)] = \frac{d}{dx} [f(x)] + \frac{d}{dx} [g(x)] = f'(x) + g'(x) The derivative of a sum of two (or more) functions is the sum of their individual derivatives. This rule allows you to differentiate a function 'term by term'. The Difference Rule \frac{d}{dx} [f(x) - g(x)] = \frac{d}{dx} [f(x)] - \frac{d}{dx} [g(x)] = f'(x) - g'(x) The derivative of a difference of two functions is the difference of their individual derivatives. This also works 'term by term'.