Power rule II

Learning Objectives Rewrite functions with radicals and rational expressions into a form suitable for the power rule. Apply the power rule to find the derivative of functions with negative integer exponents. Apply the power rule to find the derivative of functions with fractional (rational) exponents. Combine the power rule with the constant multiple and sum/difference rules to differentiate more complex functions. Differentiate multi-term functions that include terms with negative and fractional exponents. Evaluate the derivative at a specific point for functions involving negative or fractional exponents. Interpret the derivative as the slope of the tangent line for functions with non-integer exponents. How do you find the instantaneous rate of change for a function like...

TermDefinitionExample Generalized Power RuleThe rule stating that the derivative of x raised to any real number power 'n' is 'n' times x raised to the power of 'n-1'. This extends the basic power rule to include negative and fractional exponents.For f(x) = x^(1/2), the derivative f'(x) = (1/2)x^(-1/2). Negative ExponentAn exponent that indicates the reciprocal of the base raised to the positive value of the exponent.x⁻³ is equivalent to 1/x³. Fractional (Rational) ExponentAn exponent in the form of a fraction (m/n), where 'm' is the power and 'n' is the root.x^(2/3) is equivalent to the cube root of x squared, or (∛x)². Rewriting the FunctionThe critical first step of converting a function from radical or fractional form into the stan...

The Generalized Power Rule \frac{d}{dx}(x^n) = nx^{n-1}, \text{ for any real number } n This is the core rule for this lesson. To use it, bring the exponent down as a coefficient and then subtract one from the original exponent. The Constant Multiple Rule \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) When a function is multiplied by a constant, you can 'ignore' the constant, differentiate the function part, and then multiply the result by the constant. The Sum/Difference Rule \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) To differentiate a function made of several terms added or subtracted, simply differentiate each term individually and keep the plus or minus signs.