Power Rule
Find the derivative of
f(x)=4-\frac{1}{2x^2}
f'(x)=\frac{1}{x^3}
Find the deriavtive of y=frac{4}{x}-sec(x) .
frac{dy}{dx}=-frac{4}{x^2}-sec(x)tan(x)
Find the derivative of
y=frac{2-x}{x+2}
frac{dy}{dx}=frac{-4}{(x+2)^2
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
Find the equation of the tangent line to f(x)=4sin(x)-2 at
x=pi
y+2=-4(x-pi)
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}
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
Find the derivative of s(x)=x^2sin(x)
s'(x)=2xsin(x)+x^2cos(x)
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
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)
lim_{h->0}frac{(x^3)-(e^3)}{h}=
A) 0
B) 3e^2
C) e^3
D) Does not exist
B)
3e^2
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)}
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)
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}}
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})
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)
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
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
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)
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
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)}
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}
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}
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
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