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# (Solved): In Exercises 40-50, you may use the formulas for derivatives Calculate the derivatives of the funct ...

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)

.

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