y = arcsin(x/(sqrt(1 + x ^ 2))) + arccos(1/(sqrt(1 + x ^ 2))) find dy/dx

Answer 1

To find the derivative dy/dx, we first need to use the chain rule and differentiate each term separately.

Let's start with the first term y = arcsin(x/(sqrt(1 + x^2))).

Using the chain rule, we have:

dy/dx = (d/dx)arcsin(x/(sqrt(1 + x^2)))

Let u = x/(sqrt(1 + x^2)).

Differentiating u with respect to x, we have:

du/dx = (d/dx)x/(sqrt(1 + x^2))

To compute du/dx, we need to use the quotient rule:

(d/dx)x/(sqrt(1 + x^2)) = [(1)(sqrt(1 + x^2)) - x(1/2)(2x)] / (1 + x^2)

= (sqrt(1 + x^2) - x^2) / (1 + x^2)

= [(1 + x^2) - x^2] / (1 + x^2)

= 1 / (1 + x^2)

Now we can substitute du/dx into our original equation:

dy/dx = (d/dx)arcsin(u) = du/dx / sqrt(1 - u^2)

= 1 / (1 + x^2) / sqrt(1 - (x/(sqrt(1 + x^2)))^2)

= 1 / (1 + x^2) / sqrt(1 - (x^2 / (1 + x^2)))

= 1 / (1 + x^2) / sqrt((1 + x^2 - x^2) / (1 + x^2))

= 1 / (1 + x^2) / sqrt(1 / (1 + x^2))

= 1 / (1 + x^2) / (sqrt(1) / sqrt(1 + x^2))

= sqrt(1 + x^2) / (1 + x^2)

Now let's move on to the second term y = arccos(1/(sqrt(1 + x^2))).

Using the chain rule, we have:

dy/dx = (d/dx)arccos(1/(sqrt(1 + x^2)))

Let v = 1/(sqrt(1 + x^2)).

Differentiating v with respect to x, we have:

dv/dx = (d/dx)1/(sqrt(1 + x^2))

To compute dv/dx, we need to use the quotient rule:

(d/dx)1/(sqrt(1 + x^2)) = [(0)(sqrt(1 + x^2)) - 1(1/2)(2x)] / (1 + x^2)

= - x / (1 + x^2)^1.5

Now we can substitute dv/dx into our original equation:

dy/dx = (d/dx)arccos(v) = -dv/dx / sqrt(1 - v^2)

= -(- x / (1 + x^2)^1.5) / sqrt(1 - (1/(sqrt(1 + x^2)))^2)

= x / (1 + x^2)^1.5 / sqrt(1 - (1/(sqrt(1 + x^2)))^2)

= x / (1 + x^2)^1.5 / sqrt(1 - (1^2 / (1 + x^2)))

= x / (1 + x^2)^1.5 / sqrt((1 + x^2 - 1) / (1 + x^2))

= x / (1 + x^2)^1.5 / sqrt(x^2 / (1 + x^2))

= x / (1 + x^2)^1.5 / (sqrt(x^2) / sqrt(1 + x^2))

= x / (1 + x^2)^1.5 / (|x| / sqrt(1 + x^2))

= x / (1 + x^2)^1.5 / (|x| / (1 + x^2))

= x / (1 + x^2)^1.5 * ((1 + x^2) / |x|)

= x / |x|(1 + x^2)^(1.5)

Now we can simplify the expression for dy/dx as:

dy/dx = sqrt(1 + x^2) / (1 + x^2) - x / |x|(1 + x^2)^(1.5)

Simplify the expression further, we have:

dy/dx = (sqrt(1 + x^2) - x / |x|) / (1 + x^2)^(1.5)

Therefore, the derivative of y with respect to x, dy/dx, is given by:

dy/dx = (sqrt(1 + x^2) - x / |x|) / (1 + x^2)^(1.5)