Question

derivative of y=5xsin^-1 x

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the derivative of the function y = 5xsin^(-1)(x), we will use the chain rule and the derivative of arcsin(x).

The derivative of arcsin(x) is given by:
d/dx arcsin(x) = 1/√(1 - x^2)

Now, let's use the chain rule to find the derivative of y = 5xsin^(-1)(x):

dy/dx = 5 * (d/dx(x)) * sin^(-1)(x) + 5x * (d/dx(sin^(-1)(x)))

Since d/dx(x) = 1 and d/dx(sin^(-1)(x)) = 1/√(1 - x^2), we can substitute these values into the equation:

dy/dx = 5 * 1 * sin^(-1)(x) + 5x * (1/√(1 - x^2))

Simplifying the equation, we have:

dy/dx = 5sin^(-1)(x) + 5x/√(1 - x^2)

So, the derivative of y = 5xsin^(-1)(x) is 5sin^(-1)(x) + 5x/√(1 - x^2).

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the derivative of y = 5x sin⁻¹x, we will use the chain rule.

Step 1: Identify the composite function.
Let u = sin⁻¹x.
Let y = 5xu.

Step 2: Find the derivative of the composite function.
dy/du = 5x.

Step 3: Find the derivative of u with respect to x.
du/dx = 1/√(1 - x²).

Step 4: Apply the chain rule.
dy/dx = (dy/du) * (du/dx)
= (5x) * (1/√(1 - x²))
= 5x/√(1 - x²).

Therefore, the derivative of y = 5x sin⁻¹x is dy/dx = 5x/√(1 - x²).

Answer 3

[AI-generated answer — Explain Bot]

To find the derivative of y = 5xsin^(-1)(x), we can use the chain rule. The chain rule states that if y = f(g(x)), then the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x). Here, f(u) = 5u and g(x) = sin^(-1)(x).

First, let's find the derivative of g(x) = sin^(-1)(x). In order to do this, we can rewrite sin^(-1)(x) as arcsin(x). The derivative of arcsin(x) with respect to x is given by 1 / sqrt(1 - x^2).

Next, let's find the derivative of f(u) = 5u. The derivative of 5u with respect to u is simply 5.

Now, we plug these derivatives into the chain rule formula. We have dy/dx = f'(g(x)) * g'(x).

Substituting, we get dy/dx = 5 * (1 / sqrt(1 - x^2)).

Therefore, the derivative of y = 5xsin^(-1)(x) is dy/dx = 5 / sqrt(1 - x^2).