Find the derivative of the function.y=3cos4θy=3\cos^4\thetay=3cos4θ

− 12 sin ⁡ θ cos ⁡ 3 θ -12\sin\theta\cos^3\theta − 12 sin θ cos 3 θ

Find the derivative of the function. y = 3 cos 4 θ y=3\cos^4\theta y = 3 cos 4 θ

this problem, we're asked to find the derivative of the function y equals three cosine the power four of theta. Now here, because we have a lot of different things going on, we're going to have to use the chain rule because I see that I have a function inside of another function. Now, again, here whenever we have a trig function that's raised to a power, it might not be immediately clear how we're going to use the chain rule. So it can be helpful to rewrite these functions to look something more like this because we have the cosine of theta and that's raised to the power of four. So we can kind of pull that four to the outside there to make it more clear what our inside and outside functions are so that we can use the chain rule. Now here we want to find our derivative y prime. So using the chain rule here, I can see that my inside function is cosine. My outside function is this three times that cosine raised to the power of four. Now, because we're using the chain rule here, we're starting from the outside, working our way in. So here, starting with our outside function, we're going to use the power rule, pulling that four out to the front, multiplying by it. That gives me 12 times what's in those parentheses. I'm keeping that inside function as is. Remember, just treating it like it's a variable. And then raising that to the power of four minus one, which is three, again, using the power rule there. So we've found the derivative of that outside function. We are working our way inside. Now, our inside function here is that cosine. So we want to multiply by the derivative of cosine, which we know is negative sine of theta. Now we can rewrite this to match our convention by pulling that three back onto the inside because we don't need it to be out there anymore. We already found our derivative and we can move that negative sine theta to the front just so that this looks a little cleaner. So I have that negative. I have 12 sine theta, and that's multiplying cosine cubed of theta. So this is my final derivative here, y prime, negative 12 sine theta times cosine cubed of theta. Now, whenever you're rewriting or simplifying, remember to make sure that you have accounted for every single term. Now, another way that you could rewrite this or simplify it further is by using trig identities. Remember that two sine theta cosine theta is actually equal to sine of two theta. It's one of our double angle trig identities. So an equivalent statement, if you feel inclined to use your trig identities here, is going to be a negative six sine of two theta times not not squared. Sine of two theta times cosine squared of theta. I was getting ahead of myself there, but this is also an equivalent statement using trig identities to get there. Now Now if you have any questions, let us know, and I will see you in that next practice problem.

Find the derivative of the function. y = 3 cos ⁡ 4 ⁡ θ y=3\cos^4⁡\theta y = 3 cos 4 ⁡ θ

− 12 sin ⁡ θ cos ⁡ 3 θ -12\sin\theta\cos^3\theta − 12 sin θ cos 3 θ

− 4 sin ⁡ 3 θ -4\sin^3\theta − 4 sin 3 θ

− 12 sin ⁡ 3 θ -12\sin^3\theta − 12 sin 3 θ

4 sin ⁡ θ cos ⁡ 3 θ 4\sin\theta\cos^3\theta 4 sin θ cos 3 θ

12. Trigonometric Functions

Derivatives of Trig Functions

Business Calculus

12. Trigonometric Functions

Derivatives of Trig Functions

Struggling with Business Calculus?

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Multiple Choice

Find the derivative of the function.
$y=3\cos^4\theta$

$-4\sin^3\theta$

$-12\sin^3\theta$

$4\sin\theta\cos^3\theta$

$-12\sin\theta\cos^3\theta$

Verified step by step guidance

Step 1: Recognize that the given function involves a composite function. Specifically, the term 3cos⁡⁴θ involves a power of a trigonometric function, so we will need to use the chain rule to differentiate it.

Step 2: Apply the chain rule to differentiate 3cos⁡⁴θ. The chain rule states that if y = [f(g(x))]^n, then dy/dx = n[f(g(x))]^(n-1) * f'(g(x)) * g'(x). Here, f(x) = cos(θ), g(x) = θ, and n = 4.

Step 3: Differentiate the outer function first. The derivative of 3cos⁡⁴θ with respect to θ is 3 * 4 * cos³θ * (-sinθ), where -sinθ is the derivative of cosθ.

Step 4: Simplify the expression obtained in Step 3. Combine constants and terms to get -12sinθcos³θ.

Step 5: Verify the result matches the correct answer. The derivative of the function is indeed -12sinθcos³θ, as provided in the problem statement.

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