MHB Derivative of Inverse Trig Function: y=4*arcsin(x/4)

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The derivative of the function y = 4*arcsin(x/4) is derived by first expressing it in terms of sine, leading to sin(y/4) = x/4. Differentiating with respect to y gives 1/4*cos(y/4) = (1/4)*(dx/dy). Rearranging this provides dy/dx as sec(y/4). Substituting y/4 with arcsin(x/4) results in dy/dx = 4/sqrt(16 - x^2). This confirms the derivative of the inverse trigonometric function.
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Here is the question:

Derivative of Inverse Function?


Find d/dx of y = 4*arcsin(x/4)

I have posted a link there to this thread so the OP can see my work.
 
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Hello harpazo,

We are given:

$$y=4\sin^{-1}\left(\frac{x}{4} \right)$$

which implies:

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

Differentiating with respect to $y$, we obtain:

$$\frac{1}{4}\cos\left(\frac{y}{4} \right)=\frac{1}{4}\frac{dx}{dy}$$

Solving for $$\frac{dy}{dx}$$, we get:

$$\frac{dy}{dx}=\sec\left(\frac{y}{4} \right)$$

Since $$\frac{y}{4}=\sin^{-1}\left(\frac{x}{4} \right)$$ we have:

$$\frac{dy}{dx}=\sec\left(\sin^{-1}\left(\frac{x}{4} \right) \right)=\frac{4}{\sqrt{16-x^2}}$$
 
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