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

Question

𝔂 = x² sin² (2x²)

Accepted Answer

Hello. In this video, we are going to be calculating the derivative of the given function. The function given to us is Y is equal to X cubed multiplied by cosine cubed of X2. Now, in order to take the derivative of this function, we are going to need to use the product rule. We have a product of two functions. First function being X cubed, and the second function being cosine cubed of X2d. Let's go ahead and use the product rule. So, the product rule states that if we have a product of two functions, F of X and G of X, then the derivative with respect to X is the first original function F of X, multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the 1. Here, we will go ahead and label F of X as X cubed. Then we need to take this derivative. The derivative of X cube through the power rule is going to be 3 X2. So F of X is equal to 3 X2. Now we will label G of X. As cosine cubed of X squad. Now, in order to take this derivative, we are going to have to use the chain rule. Now, it's a little tricky to see at first, but when we are looking at this given function, first, let's go ahead and rewrite the function with respect to the exponent. So we can rewrite this function as cosine of X2d raised to the power of 3. So in order to take the derivative, we will go ahead and use the power rule by taking the derivative of the outermost function, which is the exponent of 3. So, the derivative of the outermost function. Is going to be 3 multiplied by cosine. X2, raise the power of 2. Then by the chain rule, we want to take the derivative of the innermost function. The derivative of the innermost function cosine X2d is also going to need to use the chain rule. For this derivative, the outermost function will be cosine, and the innermost function will be X squared. So, we will go ahead and take the derivative of cosine. The derivative of cosine is negative sign, so the derivative of the outermost function will be negative sine x2d. Then we must multiply this derivative by the innermost function, which is X2. So the derivative of X2 will be 2 X through the power rule. And this is going to be the complete chain rule of the function cosine cubed of X2d. Now let's go ahead and multiply the coefficients together. If we multiply the coefficients, that is going to leave us with -6 X. Of sign, sorry, cosine. Squared of X2 multiplied by sine of X2, and this is going to be our derivative G for the product rule. Now that we have the 4 pieces we need, we will go ahead and plug in our F of X, its derivative, and our G of X with its derivative into the product rule. That is going to leave us with X cubed. Multiplied by the quantity, negative 6 X cosine squared of X2 multiplied by sin of X2. Plus G of X, which is cosine cubed of X2, multiplied by the derivative of F of X, which is going to be 3 X2. Now, if we multiply these terms together, that is going to leave us with negative 6X to the 4th power multiplied by cosine squared of X2, multiplied by sin of X2. Plus 3 X 2 multiplied by cosine cubed of X2. Then if we were to reorder the terms, Y will equal to 3 X2, multiplied by cosine cubed of X2. Minus 6 X to the 4th power multiplied by cosine squared of X2, multiplied by sine of X2, and this is going to be the derivative Y. And with that being said, the overall solution to this problem is going to be B. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

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

𝔂 = x² sin² (2x²)

Key Concepts

Derivative

Product Rule

Chain Rule

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