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

− 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

this problem, we're asked to find the derivative of the function y equals 3 cosine the power of 4 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 a raised to the power of 4. So we can kinda pull that 4 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 wanna 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 3 times that cosine raised to the power of 4. 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 4 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 4 minuteus 1, which is 3, 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 3 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 2 sine theta cosine theta is actually equal to sine of 2 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 6 sine of 2 theta times not not squared. Sine of 2 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

− 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

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

Struggling with Calculus?

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Multiple Choice

Find the derivative of the function.
$y = 3 \cos^{4} θ$

$-4\sin^3\theta$

$-12\sin^3\theta$

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

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

Verified step by step guidance

Identify the function components: The function is a combination of trigonometric functions raised to powers and multiplied by constants. It is given as y = 3cos^4(θ) - 4sin^3(θ) - 12sin^3(θ) + 4sin(θ)cos^3(θ).

Simplify the function: Combine like terms where possible. Notice that -4sin^3(θ) and -12sin^3(θ) can be combined to -16sin^3(θ). The function simplifies to y = 3cos^4(θ) - 16sin^3(θ) + 4sin(θ)cos^3(θ).

Apply the chain rule: To find the derivative of y with respect to θ, apply the chain rule to each term. For the first term, 3cos^4(θ), use the chain rule: d/dθ[3cos^4(θ)] = 3 * 4cos^3(θ) * (-sin(θ)).

Differentiate the second term: For -16sin^3(θ), apply the chain rule: d/dθ[-16sin^3(θ)] = -16 * 3sin^2(θ) * cos(θ).

Differentiate the third term: For 4sin(θ)cos^3(θ), use the product rule: d/dθ[4sin(θ)cos^3(θ)] = 4[cos^3(θ) * cos(θ) + sin(θ) * 3cos^2(θ)(-sin(θ))].

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