Verify the differentiation formula.
d/dx[arcsin u] = u'/(sq. rt. (1-u^2))
To verify the differentiation formula, we need to find the derivative of the function arcsin(u) with respect to x. Here's a step-by-step explanation of how to approach this:
1. Start with the definition of the arcsine function, which states that arcsin(u) is the angle whose sine is u. In other words, sin(arcsin(u)) = u.
2. Let's consider u as a function of x. So, u = u(x).
3. Differentiate both sides of the equation sin(arcsin(u)) = u with respect to x:
d/dx[sin(arcsin(u))] = d/dx[u]
cos(arcsin(u)) * d/dx[arcsin(u)] = u'
4. Now, let's focus on the left side of the equation. Recall the derivative of sine function: d/dx[sin(x)] = cos(x).
5. Replace x in the derivative of sine function with arcsin(u):
d/dx[sin(arcsin(u))] = cos(arcsin(u))
6. Therefore, the left side of the equation becomes:
cos(arcsin(u))
7. Putting it all together, the equation becomes:
cos(arcsin(u)) * d/dx[arcsin(u)] = u'
8. Now, we need to find an expression for cos(arcsin(u)). By using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can solve for cos(x):
cos(x) = √(1 - sin^2(x))
9. Substitute arcsin(u) for x in the above equation:
cos(arcsin(u)) = √(1 - sin^2(arcsin(u)))
10. Remember that sin(arcsin(u)) = u. Therefore, sin^2(arcsin(u)) = u^2.
11. Simplify the expression:
cos(arcsin(u)) = √(1 - u^2)
12. Finally, substitute cos(arcsin(u)) and u' into the derived equation (from step 7):
√(1 - u^2) * d/dx[arcsin(u)] = u'
And there you have it! The differentiation formula is verified to be d/dx[arcsin(u)] = u'/(√(1 - u^2)).