Power Product and Quotient Rules

Objective: Explain power rule, apply the stated method, and check its conditions.

For a real power xⁿ on an interval where the expression is differentiable, its derivative is nxⁿ⁻¹. Constant multiples carry through differentiation, and constants differentiate to zero. A real power’s domain must still be respected. The power rule does not erase the original expression’s domain.

Objective: Explain product rule, apply the stated method, and check its conditions.

For differentiable u and v, (uv)′=u′v+uv′. Each term differentiates one factor and leaves the other unchanged. Multiplying derivatives u′v′ is not the derivative of the product in general. Expansion can provide an independent check for polynomial examples. The derivative of a product is not generally the product of derivatives.

Objective: Explain quotient rule, apply the stated method, and check its conditions.

For differentiable u,v with v≠0, (u/v)′=(u′v−uv′)/v². The order of the subtraction matters. Preserve denominator restrictions and simplify only after setting up the rule; the derivative can be checked at an allowed input. Keep the numerator order and the original nonzero-denominator condition.