Derivatives of sin^n x Calculate the following deriv...

Question

Derivatives of sin^n x Calculate the following derivatives using the Product Rule.

c. d/dx (sin⁴ x)

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with derivatives. This problem says to apply the product rule to differentiate the function H of Z, which is equal to 4 multiplied by cosine cubed of Z. We're given 4 possible choices as our answers. For choice A, we have H Z is equal to minus 12 multiplied by cosine square of Z multiplied by sine of Z. For choice B, we have H Z is equal to 12 multiplied by cosine square of Z, multiplied by sine of Z. For choice C, we have H Z is equal to 12 multiplied by cosine square of Z, and for choice D we have H Z is equal to 12 multiplied by sine square of Z. Now we need to apply the product rule to differentiate our function H of Z. So in order to use the product rule, we're going to need to rewrite our function H of Z, and this is going to be equal to or multiplied by cosine of Z, multiplied by cosine of Z. Multiplied by cosine of Z. And we also need to apply the product rules. So we call for the product rules that we take the derivative of the quantity of F multiplied by G, then that is going to be equal to F multiplied by G plus F multiplied by G. So we're gonna use this product rule to find H of Z. That means that H of Z is equal to or multiplied by the quantity, and here we'll have to apply our product rule. So if we look at our product rule, we need to take the derivative of the first term in our product, which in this case is going to be cosine of Z. You want to take a derivative of cosine of Z, I get minus sine of Z, then that gets multiplied by the second term in our product, which is cosine of Z multiplied by cosine. Of Z Then we'll have less, the first term in our product, which is cosine of Z. Multiplied by the derivative with respect to Z of the second term in our product, which is cosine of Z multiplied by cosine of Z. So, this is going to be equal to 4 multiplied by the quantity, we'll just simplify our first term here of minus cosine squared Z multiplied by sine of Z. Then we'll have plus. Cosine of Z, multiplying the quantity, and here we have to again apply our product rule, since we have two terms of cosine of Z multiplying each other. So we'll take a derivative with respect to Z of our first term, which is cosine of Z, and that could give you minus sine of Z. And that gets multiplied by our second term, which is cosine of D. Then we'll have plus our first term, which is cosine of Z, multiplied by the derivative of the second term, which is minus sine of Z. So now all I need to do is simplify this expression. So this is going to be equal to. Or multiplied by the quantity of minus cosine squared of Z multiplied by sine of Z. Then we'll need to distribute our cosine of Z into the remaining terms, and doing so for the next term, I will have minus cosine squared of Z multiplied by sine of Z. Then for the last term, I will again have minus cosine squared of Z multiplied by sine of Z. So now notice that I have 3 terms of minus cosine squared of Z multiplied by sin Z being combined, so then this is going to be equal to 4, multiplying the quantity of minus 3 multiplied by cosine squared of Z multiplied by sin of Z. And then simplifying this, this is going to be equal to -12, multiplied by cosine squared of Z, multiplied by sin of Z. So, this is the answer that we are actually looking for. And if I look at my answer choices, 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.

Derivatives of sin^n x Calculate the following derivatives using the Product Rule.
c. d/dx (sin⁴ x)

Key Concepts

Product Rule

Chain Rule

Higher Order Derivatives

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