Evaluate the derivative of the function below at the point (, ).
y=7/x + sqrt cos(x)
First, rewrite the function like this:
$\displaystyle y = 7x^{-1} + (\cos{x})^{\frac{1}{2}}$
then, by the chain rule:
$\displaystyle y^{\prime} = -7x^{-2} + \frac{1}{2}(cos{x})^{-\frac{1}{2}}(-\sin{x})$
and $\displaystyle y^{\prime}\left(\frac{\pi}{4}\right) = -11.76844$:
- $\displaystyle \frac{\pi}{4}$ is not a critical number
- the curve is falling at $\displaystyle \left(\frac{\pi}{4}, 9.754\right)$
Right. Just to clarify:
$\displaystyle
y^{\prime} = -7x^{-2} + \frac{1}{2}(cos{\frac{\pi}{4}})^{-\frac{1}{2}}(-\sin{\frac{\pi}{4}}) = -7\left(\frac{\pi}{4}\right)^{-2} + \frac{1}{2}\left(\frac{\sqrt{2}}{2}\right)^{-\frac{1}{2}}\left(-\frac{\sqrt{2}}{2}\right) = -11.7684
$