All Derivatives Review

Power Rule

100

Find the derivative of  

 f(x)=4-\frac{1}{2x^2} 

f'(x)=\frac{1}{x^3}

200

Find the derivative of  y=3\sqrt(x)-frac{6}{x^2}+5\pi^3 

frac{dy}{dx}=frac{3}{2sqrt(x)}+frac{12}{x^3}

300

 lim_{h->0}frac{(x^3)-(e^3)}{h}= 

A)  0 

B)  3e^2 

C)  e^3 

D) Does not exist

B) 

3e^2

400

A circle centered at (0,0) with radius 2 has equation  x^2+y^2=4 . What is the slope of the tangent line at  (1, sqrt(3))  ?

A)  -1  

B)  -frac{1}{sqrt(3) 

C)  frac{1}{sqrt(3) 

D)  1 

E)  sqrt(3) 

B) 

-frac{1}{sqrt(3)

500

Let  g(x)=(arccos(x^2))^5 . Then g'(x)=

A)  -10frac{(arccos(x^2))^4}{sqrt(1-x^2)} 

B)  -10frac{x(arccos(x^2))^4}{sqrt(1-x^4)} 

C)  -10frac{(arcsin(x^2))^4}{sqrt(1-x^2)} 

D)  10frac{(arccos(x^2))^4}{sqrt(1-x^2)} 

E)  10frac{(arccos(x^2))^4}{sqrt(1-x^4)} 

B) 

-10frac{x(arccos(x^2))^4}{sqrt(1-x^4)}

Trigonometry

100

Find the deriavtive of  y=frac{4}{x}-sec(x) .

frac{dy}{dx}=-frac{4}{x^2}-sec(x)tan(x)

200

 lim_{h->0}frac{tan(frac{pi}{3}+h)-tan(frac{pi}{3})}{h}= 

A)  4 

B)  2 

C)  frac{4}{3} 

D)  frac{2}{sqrt(3) 

E)  1 

A) 

4

300

 frac{d}{dx}(sin^(-1)(x)+2sqrt(x))= 

A)  -frac{1}{sin^2(x)}+frac{1}{2sqrt(x) 

B)  frac{1}{sqrt(1-x^2)}+4root(3)(x) 

C)  frac{1}{sqrt(1-x^2)}+frac{1}{sqrt(x)} 

D)  frac{1}{sqrt(x^2-1)}+4root(3)(x) 

E)  frac{1}{sqrt(x^2-1)}+frac{1}{sqrt(x)} 

C)

frac{1}{sqrt(1-x^2)}+frac{1}{sqrt(x)}

 

400

What is the slope of the line tangent to the curve  y=arctan(2x)  at the point when  x=frac{1}{2} ?

A)  frac{1}{4} 

B)  frac{1}{2} 

C)  1 

D)  2 

E)  4 

C) 

1

500

If  arctan(y)=ln(x) , then  frac{dy}{dx}= 

A)  tan(frac{1}{x}) 

B)  tan (ln(x)) 

C)  frac{1+y^2}{xy} 

D)  frac{x}{1+y^2} 

E)  frac{1+y^2}{x} 

E) 

frac{1+y^2}{x}

Product/Quotient Rules

100

Find the derivative of 

 y=frac{2-x}{x+2} 

frac{dy}{dx}=frac{-4}{(x+2)^2

200

Find the derivative of  s(x)=x^2sin(x) 

s'(x)=2xsin(x)+x^2cos(x)

300

If  f(x)=x^2ln(x) , then 

 f'(x)= 

A)  2 

B)  x+2ln(2) 

C)  2xln(x) 

D)  1+2xln(x) 

E)  x+2xln(x) 

E) 

x+2xln(x)

400

Let  f(x)=xdotg(h(x)) , where g(4)=2, g'(4)=3, h(3)=4, and h'(3)=-2. Find  f'(3) .

A)  -18 

B)  -16 

C)  -7 

D)  7 

E)  11 

B) -16

500

Find the value(s) of  frac{dy}{dx}  of  x^2y+y^2=5  at  y=1 .

A)  -frac{3}{2}  only

B)  -frac{2}{3}  only

C)  frac{2}{3}  only

D)  +-\frac{2}{3} 

E)  +-frac{3}{2} 

D) 

+-frac{2}{3}

All Deriavtives

100

If  f(x)=(1+frac{x}{20})^5 , find  f''(40) .

A)  0.068 

B)  1.350 

C)  5.400 

D)  6.750 

E)  540.000 

B) 

1.350

200

If  y=xe^x , then  frac{d^ny}{dx^n}= 

A)  e^x 

B)  e^{nx} 

C)  (x+n)e^x 

D)  x^n e^x 

E)  (x+n^2)e^x 

C) 

(x+n)e^x

300

If  f(x)=sqrt(1+sqrt(x)) , find  f'(x) .

A)  frac{-1}{4sqrt{x}sqrt{1+sqrt{x}} 

B)  frac{1}{2sqrt{x}sqrt{1+sqrt{x}} 

C)  frac{1}{4sqrt{1+sqrt{x}} 

D)  frac{1}{4sqrt{x}sqrt{1+sqrt{x}} 

E)  frac{-1}{2sqrt{x}sqrt{1+sqrt{x}} 

D) 

frac{1}{4sqrt{x}sqrt{1+sqrt{x}}

400

 frac{d}{dx}(ln(3x)5^{2x})= 

A)  frac{5^{2x}}{x}+2ln(5)ln(3x)5^(2x)  

B)  frac{5^(2x)}{3x}-2xln(3x)5^(2x) 

C)  frac{5^(2x)}{x}-ln(5)ln(3x)5^(2x)  

D)  frac{5^(2x)}{3x}+2ln(3x)5^(2x) 

E)  frac{5^(2x)}{x}+ln(5)ln(3x)5^(2x) 

A) 

frac{5^(2x)}{x}+2ln(5)ln(3x)5^(2x)

500

Find the slope of the NORMAL line to  y=xcos(xy)  at  (0,1) .

A)  1 

B)  -1 

C)  0 

D)  2 

E) Undefined

B) -1

Tangent and Normal Lines

100

Find the equation of the tangent line to  f(x)=4sin(x)-2  at 

x=pi

y+2=-4(x-pi)

200

Find the equation of the normal line to 

 y=frac{1}{2}x^2+frac{3}{4}x-4  at  x=-3 .

y+frac{7}{4}=frac{4}{9}(x+3)

300

Find the equation of the tangent line to  y=sin^-1(2x)  at the point where  x=frac{1}{4} .

y-frac{pi}{6}=frac{4}{sqrt(3)}(x-frac{1}{4})

400

Find the equation of the tangent line to  9x^2+16y^2=52 through  (2,-1) .

A)  -9x+8x-26=0 

B)  9x-8y-26=0 

C)  9x-8y-106=0 

D)  8x+9y-17=0 

E)  9x+16y-2=0 

B) 

9x-8y-26=0

500

A curve given by the equation  x^3+xy=8  has slope given by  frac{dy}{dx}=frac{-3x^2-y}{x} . The value of  frac{d^2y}{dx^2}  at the point where x=2 is

A) -6

B) -6

C) 0

D) 4

E) Undefined

C) 0