12. $$ [-/ 1.07 $$ Points $$ ] $$ DETAILS MY NOTES Find the derivative of the function. Simplify if possible. $$ y=\arccos \left(e^{6 x}\right) $$ $$ y^{\prime}=\square $$

Question

UpStudy AI · Accepted Answer

1. Let $$ u = e^{6x} $$, so that the function becomes $$ y = \arccos(u) $$.

2. The derivative of $$ \arccos(u) $$ with respect to $$ u $$ is 
   $$
   \frac{d}{du}\arccos(u) = -\frac{1}{\sqrt{1-u^2}}.
   $$

3. Next, calculate the derivative of $$ u = e^{6x} $$ with respect to $$ x $$:
   $$
   \frac{du}{dx} = 6e^{6x}.
   $$

4. Apply the chain rule:
   $$
   \frac{dy}{dx} = \frac{d}{du}\arccos(u) \cdot \frac{du}{dx} = -\frac{1}{\sqrt{1-u^2}} \cdot 6e^{6x}.
   $$

5. Substitute back $$ u = e^{6x} $$:
   $$
   \frac{dy}{dx} = -\frac{6e^{6x}}{\sqrt{1-(e^{6x})^2}}.
   $$

6. Simplify the expression by writing $$ (e^{6x})^2 $$ as $$ e^{12x} $$:
   $$
   \frac{dy}{dx} = -\frac{6e^{6x}}{\sqrt{1-e^{12x}}}.
   $$

Thus, the derivative of the function is 
$$
y' = -\frac{6e^{6x}}{\sqrt{1-e^{12x}}}.
$$
$$ y' = -\frac{6e^{6x}}{\sqrt{1-e^{12x}}} $$

Points DETAILS MY NOTES
Find the derivative of the function. Simplify if possible.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions

Points DETAILS MY NOTES Find the derivative of the function. Simplify if possible.