# Find y’ if x = 2 arccos 2t and y = 4 arcsin 2t

**Problem**

Find *y’* if *x* = 2 arccos 2*t* and *y* = 4 arcsin 2*t*.

A. 2 | C. 4 |

B. -2 | D. -4 |

**Answer Key**

**Solution**

$dy = 4 \left[ \dfrac{2 \, dt}{\sqrt{1 - (2t)^2}} \right]$

$dy = \dfrac{8 \, dt}{\sqrt{1 - 4t^2}}$

$x = 2\arccos 2t$

$dx = 2 \left[ \dfrac{-2 \, dt}{\sqrt{1 - (2t)^2}} \right]$

$dx = \dfrac{-4 \, dt}{\sqrt{1 - 4t^2}}$

$y' = \dfrac{dy}{dx}$

$y' = \dfrac{\dfrac{8 \, dt}{\sqrt{1 - 4t^2}}}{\dfrac{-4 \, dt}{\sqrt{1 - 4t^2}}}$

$y' = \dfrac{8}{-4}$

$y' = -2$ ← *answer*

**Recommended Solution**

Notice the the choices are free from any variable, which means that the answer is the same regardless of the value of *t*. You can therefore use your calculator directly by assigning a limit to its `d/dx`

function. Set the angle of your calculator into `RAD`

.

$y' = \dfrac{\dfrac{d}{dx}\big[ 4\arcsin 2x \big]_{x = 0.325}}{\dfrac{d}{dx}\big[ 2\arccos 2x \big]_{x = 0.325}}$

$y' = -2$ ← *answer*

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