Power rule I

Learning Objectives State the power rule for derivatives from memory. Apply the power rule to find the derivative of functions in the form f(x) = cx^n. Rewrite functions involving radicals and fractions into a power form suitable for differentiation. Differentiate polynomial functions by applying the power rule and sum/difference rule to each term. Correctly find the derivative of a constant function. Calculate the slope of a tangent line to a curve at a specific point using the derivative. How does a GPS calculate your car's exact speed at one single moment in time? 🚗💨 The answer lies in a powerful calculus shortcut! This tutorial introduces the Power Rule, a fundamental shortcut for finding derivatives without using the lengthy limit definition. Mastering this rule...

TermDefinitionExample DerivativeThe derivative of a function, denoted f'(x) or dy/dx, represents the instantaneous rate of change of the function at a specific point. Geometrically, it is the slope of the line tangent to the function's graph at that point.If f(x) represents the position of a car, then its derivative f'(x) represents the car's instantaneous velocity. Power FunctionA function of the form f(x) = x^n, where n is any real number (positive, negative, or a fraction).f(x) = x^3, g(x) = x^{-2}, h(x) = x^{1/2} (which is \sqrt{x}) are all power functions. CoefficientA numerical or constant quantity placed before and multiplying the variable in an algebraic expression.In the term 7x^4, the coefficient is 7. Tangent LineA straight line that 'just touches'...

The Power Rule \frac{d}{dx}(x^n) = n \cdot x^{n-1} To find the derivative of x raised to a power, bring the exponent down as a multiplier and then subtract one from the original exponent. The Constant Multiple Rule \frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x)) The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. The constant 'tags along'. The Sum/Difference Rule \frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x)) The derivative of a sum or difference of functions is the sum or difference of their individual derivatives. This allows you to differentiate polynomials term by term. The Constant Rule \frac{d}{dx}(c) = 0 The derivative of any constant number is ze...