The graphs of a function and its derivative are shown below. Label the graphs f and f', and write a short paragraph stating the criteria you used to make your selection

Use the definition of the derivative to find \displaystyle{\frac{d}{dx}}(x-x^2).

Determine \displaystyle{\frac{dy}{dx}} for each of the following functions:

y=\displaystyle{\frac{1}{3}}x^3-3x^{-5}+4\pi

Determine \displaystyle{\frac{dy}{dx}} for each of the following functions:

y=6(2x-9)^5

Determine \displaystyle{\frac{dy}{dx}} for each of the following functions:

y=\displaystyle{\frac{2}{\sqrt{x}}}+\displaystyle{\frac{x}{\sqrt{3}}}+6\sqrt[3]{x}

Determine \displaystyle{\frac{dy}{dx}} for each of the following functions:

y=\left(\displaystyle{\frac{x^2+6}{3x+4}}\right)^5

Determine \displaystyle{\frac{dy}{dx}} for each of the following functions:

y=x^2\sqrt[3]{6x^2-7}

(Simplify your answer.)

Determine \displaystyle{\frac{dy}{dx}} for each of the following functions:

y=\displaystyle{\frac{4x^5-5x^4+6x-2}{x^4}}

(Simplify your answer.)

Determine the slope of the tangent to the graph of y=(x^2+3x-2)(7-3x) at (1,8).

Determine \displaystyle{\frac{dy}{dx}} at x=-2 for y=3u^2+2u and u=\sqrt{x^2+5}.

Determine the equation of the tangent to y=(3x^{-2}-2x^3)^5 at (1,1).

The amount of pollution in a certain lake is P(t)=(t^{\displaystyle{\frac{1}{4}}}+3)^3, where t is measured in years and P is measured in parts per million (ppm). At what rate is the amount of pollution changing after 16 years?

At what point on the curve y=x^4 does the normal have a slope of 16?

Determine the points on the curve y=x^3-x^2-x+1 where the tangent is horizontal.

For what values of a and b will the parabola y=x^2+ax+b be tangent to the curve y=x^3 at point (1,1)?