Using identities Use the identity sin 2x=2 sin x cos x sin 2 t...

Question

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says using the identity cosine of 3 X is equal to 4 multiplied by cosine cubed of X minus 3 multiplied by cosine of X. Find the derivative, which is the derivative with respect to X, of the quantity of cosine of 3 X. Then use the identity sin of 3 X is equal to 3 multiplied by sin of x minus 4 multiplied by sin cubed of X to express the derivative in terms of sin of 3 X. We give them 4 possible choices as our answers. For choice A, we have the derivative with respect to X, of the quantity of cosine of 3 X is equal to -3 multiplied by sin of 3 X. For choice B, we have the derivative with respect to X of the quantity of cosine of 3 X is equal to 9 multiplied by the sin of x. For choice C, we have the derivative with respect to X of the quantity of cosine of 3 X is equal to minus 3 multiplied by sin of x. And for choice D, we have the derivative with respect to x of the quantity of cosine of 3 X is equal to 9 multiplied by sin of 3 X. Now we're asked to find the derivative with respect to X of the quantity of cosine of 3 X. Adhere to take this derivative, we're going to need to use the identity that we were provided. That means that this is going to be equal to the derivative with respect to X of the quantity of 4 multiplied by cosine cubed of X minus 3 multiplied by cosine of X. So we need to take a derivative of each of those terms, so this is going to be equal to. For our first term, we can pull the 4 out in front of the derivatives since that's the constants will have 4. And they will be taking the derivative with respect to x of cosine cube of X. So here we'll have to apply both our power rule and chain rule. So applying the power rule we'll multiply 4 by the exponent, which is 3, and that gets multiplied by cosine squared of X, and then we need to take the derivative of cosine, so we'll need to multiply this by minus sine of x. And then we'll have -3, and here we'll take a derivative of cosine X for our second term, with respect to X, and that will give us minus sine X. So we'll multiply the 3 by minus sine X. So simplifying this expression, this is going to be equal to -12, multiplied by cosine squared X, multiplied by sin of X. Plus 3 multiplied by sine of X. Now, we ultimately want everything in terms of sine, since we want to write our final answer in terms of sin of 3 X. So we're gonna replace cosine squared of X with the quantity of 1 minus sine squad of X. So this is going to be equal to -12 multiplied by the quantity of 1 minus sine squared of X in quantity multiplied by sin of X. Plus 3 multiplied by sine of X. So multiplying out the first term in our expression, this is going to be equal to. -12, sign of X. Plus 12 multiplied by sine cubed of X. Plus 3 sign of X. So combining like terms, this is going to be equal to. -9. Multiplied by sin of X. Plus 12 multiplied by sign. Cubbed of X. So factor out a -3 out of both of these terms. So this is going to be equal to -3, multiplied by the quantity of 3, sine of X. Minus 4 multiplied by sin cubed of X. Now, 3 multiplied by sin of x minus 4 multiplied by sin cubed of X was based upon our identity equal to sin of 3 X. So this is going to be equal to -3 multiplied by sin of 3 X. And so this is the answer that we are actually looking for for this problem. And so when we compare our answer to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

Key Concepts

Trigonometric Identities

Differentiation

Chain Rule

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