Find the derivatives of the functions in Exercises 1–42.

Question

𝔂 = 1 x² csc 2

2 x

Accepted Answer

Welcome back, everyone. Calculate the derivative of the function Y equals 1/3 the x cubed cotangent of 4 divided by X. So what we have to do is simply recognize that our function is a product of two functions. We have 1/3 as a constant, then we have X cubed as our first function, and our second function is cotangent of 4 divided by X. So what we're going to do is simply apply the product rule. Let's suppose that F is equal to x cubed, and our next function is G, which is cotangent of 4 divided by X. So according to the product rule, if we want to evaluate the derivative of FG, this is FG plus FG. So we need two derivatives. Let's begin with the derivative of F. Which is the derivative of Xq, and here we can apply the power rule, so it's 3 x 2, right? We have found the derivative and now we need to identify the derivative of G or G which is equal to the derivative of cotangent of 4 divided by X. And here we have a composite function. So first of all, we have to identify the derivative of cotangent with respect to the original angle 4 divided by X. Let's recall that the derivative of cotangent is negative cos squad, and we are keeping our angle the same 4 divided by x. And now according to the chain rule, we have to multiply that by the derivative of the inner function, which is the derivative of 4 divided by X, right? Or we can simply write it as 4 x raises to the power of -1. And here we can apply the power rule once again. So we're going to get -4 X to the power of -2, right? So negative of negative is going to give us a positive sign, so we can say that we're getting 4 X was the power of -2 coequant squared. Of 4 divided by X and then we can simply rewrite it as 4 divided by x quad core of 4 divided by x using the properties of exponents because we have a negative exponent, right? And now let's substitute everything into the formula. So FG derivative is equal to F which is 3x2 multiplied by G. Which is cotangent of 4 divided by X plus F which is X cubed multiplied by g derivative which is 4 divided by x2d core of 4 divided by X. And now let's simplify the result, right? So we end up with 3 X squad. Coottensions. Of where divided by X. Plus Now x cubed divided by X gives us X, so we end up with 4 x cosequent squared of 4 divided by X. And now what we have to notice is that we also have 1/3 in front, right? We only found the derivative of X cubed cotangent of 4 divided by X. So all that we have to do is simply multiply the result by 1/3, right? So if we multiply by 1/3, we're going to cancel out the 3 in front of X2, which gives us X2, cotangent. Of 4 divided by X, plus, now 4 is going to be divided by 3 because we have 1/3 in front of the whole function. So we end up with 4/3. A Coon squared off 4 divided by x. Which is essentially our final answer. To this problem. That's it. Thank you for watching.

Find the derivatives of the functions in Exercises 1–42.

𝔂 = 1 x² csc 2
2 x

Key Concepts

Derivatives

Cosecant Function (csc)

Quotient Rule

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