Find $\frac{d y}{d x}$
\[
y=\cot x-6 \sqrt{x}+\frac{8}{e^{x}}
\]

Answer

Expert–verified

Hide Steps

Answer

Thus, the derivative of the given function is $\boxed{-\csc^2 x -3x^{-\frac{1}{2}} -\frac{8}{e^x}}$.

Steps

Step 1 ：First, we need to find the derivative of each term in the equation separately.

Step 2 ：For the first term, $y = \cot x$, the derivative is $-\csc^2 x$.

Step 3 ：For the second term, $y = -6 \sqrt{x}$, the derivative is $-3x^{-\frac{1}{2}}$.

Step 4 ：For the third term, $y = \frac{8}{e^x}$, the derivative is $-\frac{8}{e^x}$.

Step 5 ：Adding these derivatives together, we get $\frac{d y}{d x} = -\csc^2 x -3x^{-\frac{1}{2}} -\frac{8}{e^x}$.

Step 6 ：Thus, the derivative of the given function is $\boxed{-\csc^2 x -3x^{-\frac{1}{2}} -\frac{8}{e^x}}$.