3.3 Derivatives- Power Rule, Product & Quotient

The Power Rule states that if $$f(x) = x^n$$ , then the derivative $$f'(x) = n*x^{n-1}$$ .

Use the Product Rule: If $$u(x)$$ and $$v(x)$$ are functions, then $$(uv)' = u'v + uv'$$ .

Use the Quotient Rule: If $$u(x)$$ and $$v(x)$$ are functions, then $$(u/v)' = (u'v - uv')/v^2$$ .

The derivative is $$f'(x) = 4$$ .

The derivative is $$y' = -\frac{5}{(x-3)^2}$$ .

Yes, differentiating a function refers to the process of finding its derivative.

To find the tangent line at $$x = a$$ , calculate the derivative at that point to find the slope, then use the point-slope form of a line.

The derivative is $$y' = \frac{-52x}{(2x^2 - 5)^2}$$ .

A critical point is where the derivative is zero or undefined, indicating potential local maxima or minima.

The first derivative test helps determine whether a critical point is a local maximum, minimum, or neither.