What is the derivative of y = arcsin ( 3 x 4 ) ?

Question

What is the derivative of       y    =          arcsin              (                              3            x                    4                )            ?

awesomeamber802x · Accepted Answer

Use the Chain rule and the definition of the derivative of the arcsin of a function.
First, understand that

y = \sin^{- 1} (3 \frac{x}{4})

is way of saying

y = f (g (x))

Second, let

f (m) = \sin^{- 1} (m)

(i.e., the "outside" part) and

m = g (x) = 3 \frac{x}{4}

(i.e., the "inside" part).
Third, use the Chain Rule which states

\frac{d y}{d x} = \frac{d y}{d m} \cdot \frac{d m}{d x} = f' (m) g' (x)

Fourth, find

\frac{d y}{d m}

\frac{d y}{d m} = f' (m) = \frac{d}{d m} (\sin^{- 1} (m)) = \frac{1}{\sqrt{1 - x^{2}}}

Fifth, find

\frac{d m}{d x}

\frac{d m}{d x} = g' (x) = \frac{d}{d x} (3 \frac{x}{4}) = \frac{3}{4}

Hence, multiply your results:

\frac{d y}{d x} = \frac{d y}{d m} \frac{d m}{d x} = \frac{1}{\sqrt{1 - x^{2}}} \cdot \frac{3}{4} = \frac{3}{4 \sqrt{1 - x^{2}}}

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What is the derivative of y=arcsin((3x)/4)?

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