Find the Derivative - d/dx y=arcsin(6x+1)

Problem

d()/d(x)*arcsin(6*x+1)

Solution

1. Identify the outer function as arcsin(u) and the inner function as u=6*x+1

2. Apply the chain rule which states that d(arcsin(u))/d(x)=1/√(,1−u2)⋅d(u)/d(x)

3. Differentiate the inner function u=6*x+1 with respect to x to get d(u)/d(x)=6

4. Substitute u and d(u)/d(x) back into the chain rule formula.

d(arcsin(6*x+1))/d(x)=1/√(,1−(6*x+1)2)⋅6

5. Expand the expression inside the square root.

(6*x+1)2=36*x2+12*x+1

6. Simplify the radicand by subtracting the expanded expression from 1

1−(36*x2+12*x+1)=1−36*x2−12*x−1

1−36*x2−12*x−1=−36*x2−12*x

7. Combine all parts into the final derivative form.

d(arcsin(6*x+1))/d(x)=6/√(,−36*x2−12*x)

Final Answer

d(arcsin(6*x+1))/d(x)=6/√(,−36*x2−12*x)

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