11. [3/5 Points] DETAILS MY NOTES SCALCET9 3.XP.6.037.MI.SA. PREVIOUS ANSWERS ASK YOUR TEACHER PRACTICE ANOTHER This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise Use logarithmic differentiation to find the derivative of the function. $$ y=x^{7 \cos (x)} $$ Step 1 Using properties of logarithms, we can rewrite the equation as $$ \ln (y)=\ln \left(x^{7} \cos (x)\right. $$, which is equivalent to $$ \ln (y)=(7 \cos (x), 7 \cos (x) \quad(\ln (x)) $$ Step 2 Note that $$ 7 \cos (x) \ln (x) $$ is a product. Therefore, its derivative is given by $$ (7 \cos (x))\left(\begin{array}{|c} \frac{1}{x} \ \frac{1}{x} \ \hline \end{array}\right)+(\ln (x))(-7 \sin (x) $$ Step 3 On the other hand, the derivative of $$ \ln (y) $$ is $$ x^{7 \cos (x)}\left(\frac{7 \cos (x)}{x}-7 \ln (x) \sin (x)\right) \times y^{\prime} $$ SUBMIT SKIP (YOU CANNOT COME BACK)

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Let's go through the steps of logarithmic differentiation for the function $$ y = x^{7 \cos (x)} $$.

### Step 1
We start by taking the natural logarithm of both sides:

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

Using the properties of logarithms, we can rewrite this as:

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

### Step 2
Next, we differentiate both sides with respect to $$ x $$. The left side becomes:

$$
\frac{1}{y} \frac{dy}{dx}
$$

For the right side, we need to use the product rule on $$ 7 \cos(x) \ln(x) $$:

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

So, we have:

$$
\frac{1}{y} \frac{dy}{dx} = 7 \cos(x) \cdot \frac{1}{x} - 7 \ln(x) \sin(x)
$$

### Step 3
Now, we can express the derivative $$ \frac{dy}{dx} $$:

$$
\frac{dy}{dx} = y \left(7 \cos(x) \cdot \frac{1}{x} - 7 \ln(x) \sin(x)\right)
$$

Substituting back $$ y = x^{7 \cos(x)} $$:

$$
\frac{dy}{dx} = x^{7 \cos(x)} \left(7 \cos(x) \cdot \frac{1}{x} - 7 \ln(x) \sin(x)\right)
$$

This is the derivative of the function $$ y = x^{7 \cos(x)} $$ using logarithmic differentiation.

If you need further assistance or calculations, feel free to ask!
The derivative of $$ y = x^{7 \cos(x)} $$ is: $$ \frac{dy}{dx} = x^{7 \cos(x)} \left(7 \cos(x) \cdot \frac{1}{x} - 7 \ln(x) \sin(x)\right) $$

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