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In Exercises 40-50, you may use the formulas for derivatives Calculate the derivatives of the functions in Exercises 34-39 using the General Power Rule. Where is each derivative valid?

`f(x)=x^(-17)`

`g(t)=t^(22)`

`y=x^((1)/(3))`

`y=x^(-(1)/(3))`

`t^(-2.25)`

`s^((119)/(4))`

In Exercises 1-12, find an equation of the straight line tangent to the given curve at the point indicated.

`y=3x-1`

at

`(1,2)`

`y=(x)/(2)`

at

`(a,(a)/(2))`

`y=2x^(2)-5`

at

`(2,3)`

`y=6-x-x^(2)`

at

`x=-2`

`y=x^(3)+8`

at

`x=-2`

`y=(1)/(x^(2)+1)`

at

`(0,1)`

`y=\sqrt(x+1)`

at

`x=3`

`y=(1)/(\sqrt(x))`

at

`x=9`

`y=(2x)/(x+2)`

at

`x=2`

`y=\sqrt(5-x^(2))`

at

`x=1`

`y=x^(2)`

at

`x=x_(0)`

`y=(1)/(x)`

at

`(a,(1)/(a))`

Find all points on the curve

`y=x^(3)-x+1`

where the tangent line is parallel to the line

`y=2x+5`

. Find all points on the curve

`y=(1)/(x)`

where the tangent line is perpendicular to the line

`y=4x-3`

. For what value of the constant

`k`

is the line

`x+y=k`

normal to the curve

`y=x^(2)`

? For what value of the constant

`k`

do the curves

`y=kx^(2)`

and

`y=k(x-2)^(2)`

intersect at right angles? Hint: Where do the curves intersect? What are their slopes there? Use a graphics utility to plot the following curves. Where does the curve have a horizontal tangent? Does the curve fail to have a tangent line anywhere?

`y=x^(3)(5-x)^(2)`

`y=2x^(3)-3x^(2)-12x+1`

established in this section. Calculate

`(d)/(ds)\sqrt(s)|_(s)=9`

. Find

`F^(')((1)/(4))`

if

`F(x)=(1)/(x)`

. Find

`f^(')(8)`

if

`f(x)=x^(-(2)/(3))`

. Find

`d(y)/(d)t|_(t)=4`

if

`y=t^((1)/(4))`

. Find an equation of the straight line tangent to the curve

`y=\sqrt(x)`

at

`x=x_(0)`

. Find an equation of the straight line normal to the curve

`y=(1)/(x)`

at the point where

`x=a`

. Show that the curve

`y=x^(2)`

and the straight line

`x+4y=18`

intersect at right angles at one of their two intersection points. Hint: Find the product of their slopes at their intersection points. There are two distinct straight lines that pass through the point

`(1,-3)`

and are tangent to the curve

`y=x^(2)`

. Find their equations. Hint: Draw a sketch. The points of tangency are not given; let them be denoted

`(a,a^(2))`

. Find equations of two straight lines that have slope -2 and are tangent to the graph of

`y=(1)/(x)`

. Find the slope of a straight line that passes through the point

`(-2,0)`

and is tangent to the curve

`y=\sqrt(x)`

.