`y = 25arcsin(x/5) -xsqrt(25-x^2)` Find the derivative of the function

`y=25arcsin(x/5) - xsqrt(25-x^2)`

Before taking the derivative, express the radical in exponent form.

`y=25arcsin(x/5) - x(25-x^2)^(1/2)`

To get y', take the derivative of each term.

`y' = d/dx[25arcsin(x/5)] - d/dx[x(25-x^2)^(1/2)]`

`y' = 25d/dx[arcsin(x/5)] - d/dx[x(25-x^2)^(1/2)]`

Take note that the derivative formula of arcsine is `d/dx[arcsin(u)] = 1/sqrt(1-u^2)*(du)/dx` .

Applying that formula, y' will become:

`y'=25* 1/sqrt(1-(x/5)^2) *d/dx(x/5) - d/dx[x(25-x^2)^(1/2)]`

`y'=25* 1/sqrt(1-(x/5)^2) *1/5 - d/dx[x(25-x^2)^(1/2)]`

`y'=25* 1/sqrt(1- x^2/25)*1/5 - d/dx[x(25-x^2)^(1/2)]`

`y'=25* 1/((1/5)sqrt(25- x^2))*1/5 - d/dx[x(25-x^2)^(1/2)]`

`y'=25/sqrt(25-x^2)...

`y=25arcsin(x/5) - xsqrt(25-x^2)`

Before taking the derivative, express the radical in exponent form.

`y=25arcsin(x/5) - x(25-x^2)^(1/2)`

To get y', take the derivative of each term.

`y' = d/dx[25arcsin(x/5)] - d/dx[x(25-x^2)^(1/2)]`

`y' = 25d/dx[arcsin(x/5)] - d/dx[x(25-x^2)^(1/2)]`

Take note that the derivative formula of arcsine is `d/dx[arcsin(u)] = 1/sqrt(1-u^2)*(du)/dx` .

Applying that formula, y' will become:

`y'=25* 1/sqrt(1-(x/5)^2) *d/dx(x/5) - d/dx[x(25-x^2)^(1/2)]`

`y'=25* 1/sqrt(1-(x/5)^2) *1/5 - d/dx[x(25-x^2)^(1/2)]`

`y'=25* 1/sqrt(1- x^2/25)*1/5 - d/dx[x(25-x^2)^(1/2)]`

`y'=25* 1/((1/5)sqrt(25- x^2))*1/5 - d/dx[x(25-x^2)^(1/2)]`

`y'=25/sqrt(25-x^2) -d/dx[x(25-x^2)^(1/2)]`

To take the derivative of the second term, apply the product rule `d/dx(u*v) = u*(dv)/dx + v*(du)/dx` .

Applying this, the y' will be:

`y'=25/sqrt(25-x^2) - [x*d/dx((25-x^2)^(1/2)) + (25-x^2)^(1/2)*d/dx(x)]`

Also, use the derivative formula `d/dx(u^n) = n*u^(n-1)*(du)/dx` .

`y'=25/sqrt(25-x^2) - [x*1/2*(25-x^2)^(-1/2)*d/dx(25-x^2) + (25-x^2)^(1/2)*1]`

`y'=25/sqrt(25-x^2) - [x*1/2*(25-x^2)^(-1/2)*(-2x) + (25-x^2)^(1/2)*1]`

`y'=25/sqrt(25-x^2) - [-x^2(25-x^2)^(-1/2) + (25-x^2)^(1/2)]`

Then, express this with positive exponent only.

`y'=25/sqrt(25-x^2) - [-x^2/(25-x^2)^(1/2) + (25-x^2)^(1/2)]`

Also, convert the fractional exponent to radical form.

`y'=25/sqrt(25-x^2) - [-x^2/sqrt(25-x^2) + sqrt(25-x^2)]`

So the derivative of the function simplifies to:

`y'=25/sqrt(25-x^2) +x^2/sqrt(25-x^2) - sqrt(25-x^2)`

`y'= (x^2+25)/sqrt(25 - x^2) - sqrt(25-x^2)`

`y'= (x^2+25)/sqrt(25 - x^2) - sqrt(25-x^2)/1* sqrt(25-x^2)/sqrt(25-x^2)`

`y'= (x^2+25)/sqrt(25 - x^2)-(25-x^2)/sqrt(25-x^2)`

`y'= (x^2+25 - (25-x^2))/sqrt(25-x^2)`

`y'=(2x^2)/sqrt(25-x^2)`

Therefore, the derivative of the function is  `y'=(2x^2)/sqrt(25-x^2)` .