Solve the problem: 10) Use the trigonometric identity tan 2x = 2 tan X to find Dxtan 2x: 1 tan2x

Solve the problem: 10) Use the trigonometric identity tan 2x = 2 tan X to find Dxtan 2x: 1 tan2x

Solve the problem: 10) Use the trigonometric identity tan 2x = 2 tan X to find Dxtan 2x: 1 tan2x

Expert's Answer

In this problem we are given the trignometric identity than 2 x is equal to 2 tan x, divided by 1 minus tan square x. We are asked to use this identity to find the derivative dx of tan 2 x. Now the derivative dx of tan 2 x is using this identity. The derivative d by d x of 2 tan x, divided by 1 minus tan square x. Now this derivative has to be evaluated using the quotient rule of derivatives as the denominator: 1 minus 10 square x times. The derivative d by d x of the numerator 2 tan x minus the numerator 2 than x times the derivative d by d x of the denominator: 1 minus tan square x, divided by the denominator square, which is 1 minus tan square x. The whole square now evaluating we get. This is 1 minus tan square x times the derivative of 2 tnistwoc square x, minus 2 tan x times. The derivative of 1 minus tan square x using chain rule is negative, 210 x times 6 square x. The whole divided by 1 minus tan square x, the whole square. Now this simplifies to 2 c square x, minus 2 c square x, tan square x, plus 46 square x, tan square x, the whole divided by 1 minus tan square x, the whole square. Now this further simplifies to 2 c square x, plus 2 c square x, tan square x, the whole divided by 1 minus 10 square x. The whole square- that is this, can be also written as 2 c square x times, 1 plus tan square x. The whole divided by 1 minus tan square x the whole square. Now we know that 1 plus tan square x square x. So this is s square x, times 6 square x, divided by 1 minus 10 square x the whole square. Now this can be written as 2 times c x, divided by c square x, divided by 1 minus 10 square x, the whole square. Now we have c square x, divided by 1. Minus tan square x can be written as 1 by co square x. Since c square x is 1 by cos. Square x, divided by 1, minus sine tan square x can be written as sine squared x by cos square x. Now, upon simplification, we get. This is 1 by cos square x times cos square x, divided by cos square x, minus sine square x. Now from here we get. This is 1 divided by cos square x, minus sine square x now cos square x by sine square x is cos 2 x. So that this is 1 divided by cos 2 x, which can be also written as c co, 2 x, so that we can write this as 2 times c 2 x. The whole square, which is 2 times c square 2 x, which is the derivative of tan 2 x, and this is the required answer.

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