Find the Derivative - d/dx cos(2arctan(x))

Problem

d(cos(2*arctan(x)))/d(x)

Solution

1. Identify the outer function and the inner function to apply the chain rule.

2. Apply the chain rule by differentiating the outer function cos(u) with respect to u=2*arctan(x)

d(cos(2*arctan(x)))/d(x)=−sin(2*arctan(x))⋅(d(2)*arctan(x))/d(x)

3. Differentiate the inner expression 2*arctan(x) using the constant multiple rule and the derivative of arctan(x)

(d(2)*arctan(x))/d(x)=2/(1+x2)

4. Substitute the inner derivative back into the chain rule expression.

d(cos(2*arctan(x)))/d(x)=−sin(2*arctan(x))⋅2/(1+x2)

5. Simplify the trigonometric expression using the double-angle identity sin(2*θ)=2*sin(θ)*cos(θ) where θ=arctan(x)

sin(2*arctan(x))=2*sin(arctan(x))*cos(arctan(x))

6. Evaluate the trigonometric functions of arctan(x) using a right triangle where the opposite side is x and the adjacent side is 1 making the hypotenuse √(,1+x2)

sin(arctan(x))=x/√(,1+x2)

cos(arctan(x))=1/√(,1+x2)

7. Combine these values to simplify the sine double-angle term.

sin(2*arctan(x))=2⋅x/√(,1+x2)⋅1/√(,1+x2)=(2*x)/(1+x2)

8. Multiply the simplified terms to find the final derivative.

d(cos(2*arctan(x)))/d(x)=−(2*x)/(1+x2)⋅2/(1+x2)

Final Answer

d(cos(2*arctan(x)))/d(x)=−(4*x)/((1+x2)2)

Want more problems? Check here!