Use logarithmic differentiation to find the derivative of the function. $$ y=x^{7 \cos (x)} $$ Step 1 Using properties of logarthms, we can rewrite the equation as $$ \ln (y)=\ln \left(x^{7} \cos (x)\right) $$, which is equivalent $$ \ln (y)=(7 \cos (x), 7 \cos (x) $$ Step 2 Note that $$ 7 \cos (x) \ln (x) $$ is a product. Therefore, its derivative is given by $$ (7 \cos (x))\left(\frac{1}{x}, \frac{1}{x}\right)+(\ln (x))(-7 \sin (x),-7 \sin (x)) $$ Step 3 On the other hand, the derivative of $$ \ln (y) $$ is Step 4 Equating the two derivatives, we have $$ \frac{y^{\prime}}{y}=\frac{7 \cos (x)}{x}-7 \ln (x) \sin (x) $$ If we replace $$ y $$ with $$ x^{7 \cos (x)} $$ and solve for $$ y^{\prime} $$, we can conclude that the derivative is $$ y^{\prime}=\square $$ SUBMIT SKIP (YOU CANNOT COME BACK)

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**Step 1. Take the natural logarithm of both sides**

$$
y = x^{7\cos(x)}
$$
Taking the natural logarithm on both sides:

$$
\ln(y) = \ln\left(x^{7\cos(x)}\right)
$$

Using the logarithm property $$\ln(a^b) = b \ln(a)$$, we have:

$$
\ln(y) = 7\cos(x)\ln(x)
$$

**Step 2. Differentiate both sides with respect to $$x$$**

Differentiate the left-hand side using the chain rule:

$$
\frac{d}{dx} \ln(y) = \frac{y'}{y}
$$

Differentiate the right-hand side, which is a product $$7\cos(x)\ln(x)$$. Use the product rule:

$$
\frac{d}{dx} \left[7\cos(x)\ln(x)\right] = 7\cos(x)\frac{d}{dx}[\ln(x)] + 7\ln(x)\frac{d}{dx}[\cos(x)]
$$

Compute the derivatives:

$$
\frac{d}{dx}[\ln(x)] = \frac{1}{x}, \quad \frac{d}{dx}[\cos(x)] = -\sin(x)
$$

So the derivative becomes:

$$
7\cos(x)\left(\frac{1}{x}\right) + 7\ln(x)\left(-\sin(x)\right) = \frac{7\cos(x)}{x} - 7\ln(x)\sin(x)
$$

**Step 3. Equate the derivatives**

We now have:

$$
\frac{y'}{y} = \frac{7\cos(x)}{x} - 7\ln(x)\sin(x)
$$

**Step 4. Solve for $$y'$$**

Multiply both sides by $$y$$:

$$
y' = y\left(\frac{7\cos(x)}{x} - 7\ln(x)\sin(x)\right)
$$

Recall that $$y = x^{7\cos(x)}$$. Substituting back gives:

$$
y' = x^{7\cos(x)}\left(\frac{7\cos(x)}{x} - 7\ln(x)\sin(x)\right)
$$
The derivative of $$ y = x^{7\cos(x)} $$ is: $$ y' = x^{7\cos(x)} \left( \frac{7\cos(x)}{x} - 7\ln(x)\sin(x) \right) $$

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