Question

Cos^7*8x найти производную.

Answer 1

(cos^7(8x))'=7cos^6(8x)*(cos(8x))'=
7cos^6(8x)*(-sin(8x))*(8x)'=
-56cos^6(8x)*sin8x

Answer 2

To find the derivative of the function $y = \cos^7(8x)$ with respect to $x$, we'll need to use the chain rule.

The chain rule states that if we have a composition of functions $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}$.

Let's apply the chain rule step by step:

Step 1: Find the derivative of $y = \cos^7(8x)$ with respect to $\cos(8x)$. Let $u = \cos(8x)$. $y = u^7$ $\frac{dy}{du} = 7u^{7-1} = 7u^6$

Step 2: Find the derivative of $u = \cos(8x)$ with respect to $x$. Using the chain rule again, let $v = 8x$. $u = \cos(v)$ $\frac{du}{dv} = -\sin(v) = -\sin(8x)$

Step 3: Find the derivative of $v = 8x$ with respect to $x$. $v' = 8$

Step 4: Combine the derivatives using the chain rule. $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} = 7u^6 \cdot (-\sin(8x)) \cdot 8$

Now, substitute back $u = \cos(8x)$: $\frac{dy}{dx} = 7\cos^6(8x) \cdot (-\sin(8x)) \cdot 8$

Simplifying: $\frac{dy}{dx} = -56\cos^6(8x)\sin(8x)$

So, the derivative of $\cos^7(8x)$ with respect to $x$ is $-56\cos^6(8x)\sin(8x)$.