Find the Derivative - d/dx y=7arctan(x- square root of 1+x^2)

Problem

d()/d(x)*7*arctan(x−√(,1+x2))

Solution

1. Identify the function and the constant multiple rule. The constant 7 can be moved outside the derivative.

d(y)/d(x)=7d(arctan(x−√(,1+x2)))/d(x)

2. Apply the chain rule for the arctangent function, where d(arctan(u))/d(x)=1/(1+u2)⋅d(u)/d(x) Let u=x−√(,1+x2)

d(y)/d(x)=7⋅1/(1+(x−√(,1+x2))2)⋅d(x−√(,1+x2))/d(x)

3. Differentiate the inner expression x−√(,1+x2) using the power rule and chain rule.

d(x−√(,1+x2))/d(x)=1−1/(2√(,1+x2))⋅2*x

d(x−√(,1+x2))/d(x)=1−x/√(,1+x2)

4. Expand the denominator 1+(x−√(,1+x2))2 in the main expression.

1+(x2−2*x√(,1+x2)+1+x2)

2+2*x2−2*x√(,1+x2)

2*(1+x2−x√(,1+x2))

5. Substitute the results back into the derivative formula.

d(y)/d(x)=7⋅1/(2*(1+x2−x√(,1+x2)))⋅(1−x/√(,1+x2))

6. Simplify the second factor by finding a common denominator.

1−x/√(,1+x2)=(√(,1+x2)−x)/√(,1+x2)

7. Combine the terms and simplify the expression. Notice that 1+x2−x√(,1+x2)=√(,1+x2)*(√(,1+x2)−x)

d(y)/d(x)=7/(2√(,1+x2)*(√(,1+x2)−x))⋅(√(,1+x2)−x)/√(,1+x2)

d(y)/d(x)=7/(2*(√(,1+x2))2)

d(y)/d(x)=7/(2*(1+x2))

Final Answer

d()/d(x)*7*arctan(x−√(,1+x2))=7/(2*(1+x2))

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