Let y=2e^cos(x)
A.) Calculate dy/dx and d^2 y/ dx^2
Show all the work.
If anyone can help me with this problem it says to show all the work thankyou!!! very urgent help pleaseee!!!!
Are you saying you have to do this from the limit definition? In terms of the "shortcuts" the only way you can do this problem is using the chain rule. You must have covered that at some point.
$\displaystyle y = 2e^{cos(x)}$
$\displaystyle \frac{dy}{dx} = 2 e^{cos(x)} \cdot -sin(x)$
$\displaystyle \frac{dy}{dx} = -2~sin(x)~e^{cos(x)}$
Now use the product rule and chain rule again to find the second derivative.
-Dan
The product rule is, given $\displaystyle h(x) = f(x) \cdot g(x)$ then $\displaystyle h^{\prime}(x) = f^{\prime}(x) \cdot g(x) + f(x) \cdot g^{\prime}(x)$
The first term looks good, but the second term has some problems.
$\displaystyle \frac{dy}{dx} = -2~sin(x)~e^{cos(x)}$
Let $\displaystyle f(x) = sin(x)$ and $\displaystyle g(x) = e^{cos(x)}$ Then $\displaystyle f^{\prime}(x) = cos(x)$ and $\displaystyle g^{\prime}(x) = -sin(x)~e^{cos(x)}$. Now put these together.
-Dan