Find the Derivative - d/dx y=arcsin(2x+1)

Problem

d()/d(x)*arcsin(2*x+1)

Solution

1. Identify the outer function as arcsin(u) and the inner function as u=2*x+1

2. Apply the chain rule, which states that d(y)/d(x)=d(y)/d(u)⋅d(u)/d(x)

3. Differentiate the outer function using the rule d(arcsin(u))/d(u)=1/√(,1−u2)

4. Differentiate the inner function u=2*x+1 to get d(u)/d(x)=2

5. Substitute the expressions back into the chain rule formula.

d(y)/d(x)=1/√(,1−(2*x+1)2)⋅2

6. Expand the binomial (2*x+1)2 inside the square root.

(2*x+1)2=4*x2+4*x+1

7. Simplify the expression inside the square root by subtracting the expanded binomial from 1.

1−(4*x2+4*x+1)=−4*x2−4*x

8. Factor the denominator to simplify further if possible.

√(,−4*(x2+x))=2√(,−x2−x)

9. Cancel the common factor of 2 in the numerator and denominator.

2/(2√(,−x2−x))=1/√(,−x2−x)

Final Answer

d(arcsin(2*x+1))/d(x)=1/√(,−x2−x)

Want more problems? Check here!