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Find derivatives of inverse trigonometric functions

14y^2-21x=a(2y^2-3x)

a

is a constant, what is the value of

a

\frac{1}{4}

inch cubes does it take to fill a box with width

3\frac{1}{4}

4\frac{1}{2}

inches, and height

2\frac{3}{4}

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2

574

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40

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644

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1

536

\begin{array}{l} h(x)=\arcsin (4 x) \\ h^{\prime}(x)=? \end{array}

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1

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\frac{1}{\sqrt{1-16 x^{2}}}

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\frac{4}{\sqrt{1-16 x^{2}}}

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\frac{1}{\sqrt{1+16 x^{2}}}

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\frac{4}{\sqrt{1+16 x^{2}}}

g(x)=\arccos (2 x)

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g^{\prime}(x)=?

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1

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\frac{-2}{\sqrt{1-4 x^{2}}}

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\frac{2}{\sqrt{1-4 x^{2}}}

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\frac{-2}{\sqrt{4 x^{2}-1}}

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\frac{2}{\sqrt{4 x^{2}-1}}

\frac{d}{d x}\left[\arcsin \left(\frac{x}{6}\right)\right]

x=4

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Use an exact expression.

f(x)=\arctan (3 x)

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f^{\prime}(x)=

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1

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\frac{3}{1-9 x^{2}}

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\frac{-3}{1-9 x^{2}}

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\frac{3}{1+9 x^{2}}

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\frac{-3}{1+9 x^{2}}

y=\arcsin (-4 x)

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\frac{d y}{d x}

x=-\frac{1}{6}

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Use an exact expression.

y=\arcsin \left(\frac{x}{4}\right)

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\frac{d y}{d x}=?

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1

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\frac{1}{\sqrt{1-\frac{x^{2}}{16}}}

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\frac{1}{4 \sqrt{1-\frac{x^{2}}{16}}}

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\frac{1}{\sqrt{1-\frac{x^{2}}{4}}}

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\frac{1}{4 \sqrt{1-\frac{x^{2}}{4}}}

\frac{d}{d x}[\arctan (2 x)]

x=-2

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Use an exact expression.

y=\arccos \left(-\frac{x}{5}\right)

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\frac{d y}{d x}

x=2

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Use an exact expression.

y=\arctan (-4 x)

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\frac{d y}{d x}

x=3

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Use an exact expression.

\frac{d}{d x}[\arccos (-2 x)]

x=\frac{1}{4}

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Use an exact expression.

y=\arctan (-6 x)

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\frac{d y}{d x}=?

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1

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\frac{1}{-6\left(1+36 x^{2}\right)}

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\frac{1}{-6(1-6 x)}

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\frac{-6}{1+36 x^{2}}

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\frac{-6}{1-6 x}

46

.) The graph of

f

includes the point

(2,6)

and the slope of the tangent line to

f

x

is given by the expression

3 x+4

f(-2)

Find the volume of the given solid,

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bounded by the cylinder

y^{2}+z^{2}=64

x=2 y, x=0, z=0

in the first octant

\frac{dy}{dx}

for the given function.

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\frac{dy}{dx}=\square=x^{2}-\csc(x)+5

Find the derivative of

y

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y=3t^{3}\tanh\left(\frac{1}{t^{2}}\right)

f(t) = 14(0.85)^t

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The function models

f

, the value of a professional quality video camera, in thousands of dollars,

t

years after its purchase. What is the value of the camera at the time of purchase?

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1

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\$11,900

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\$14,000

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\$15,000

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\$85,000

Find the derivative of the following function.

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y=\ln \left(9 x^{6}+4 x^{5}\right)

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y^{\prime}=

Kofi tried to solve the differential equation

\frac{d y}{d x}=3 x^{-2} e^{-y}

. This is his work:

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\frac{d y}{d x}=3 x^{-2} e^{-y}

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1

\quad \int e^{y} d y=\int 3 x^{-2} d x

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2

\quad e^{y}=-\frac{3}{x}

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3

\quad \ln \left(e^{y}\right)=\ln \left(-\frac{3}{x}\right)

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4

\quad y=\ln \left(-\frac{3}{x}\right)+C

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Is Kofi's work correct? If not, what is his mistake?

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1

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(A) Kofi's work is correct.

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1

is incorrect. The separation of variables wasn't done correctly.

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2

is incorrect. The right-hand side of the equation should be

-\frac{3}{x}+C

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3

is incorrect. Kofi should not have a negative symbol inside of a natural logarithm.

Solve the equation.

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\frac{d y}{d x}=\frac{\cos (x)}{\sin (y)}

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1

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y=\arcsin (-\sin (x))+C

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y=\arcsin (-\sin (x)+C)

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y=\arccos (-\sin (x))+C

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y=\arccos (-\sin (x)+C)

Find the derivative of

f(x)

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f(x) = \tan^{-1}{x}

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f'(x) =