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)
?