lecture 13 even more diff

For $f(x)=\cos (\frac{2}{3}x)$, find $f^{\prime }(\frac{7\pi }{4})$.

$f^{\prime }(x)=\frac{-2}{3}\sin (\frac{2}{3}x)$
$f^{\prime }(\frac{7\pi }{4})=\frac{-2}{3}\cdot \frac{-1}{2}=\frac{1}{3}$

For $f(x)=\cos (3x^2+4x+2)$, find $f^{\prime \prime }(x)$.

$f^{\prime }(x)=(3x+4)(-\sin (3x^2+4x+2))$
$f^{\prime \prime }(x)=(3)(-\sin (3x^2+4x+2))+(3x+4)(-\cos (3x^2+4x+2)((3x+4)))$

A tangent to the graph of $y=\ln (2x)$ has a gradient of 3. This tangent will cross the y−axis at:

$\frac{dy}{dx}=3=\frac{2}{2x}$
$6x=2$
$x=\frac{1}{3}$

$(\frac{1}{3},\ln (\frac{2}{3}))$

$y-y_1=m(x-x_1)$

$y-\ln (\frac{2}{3})=3(x-\frac{1}{3})$
$y=3x-1+\ln (\frac{2}{3})$

Owen is participating in a sports competition involving running and rowing. When he is rowing he can go across the circular lake however when he is running he can only go around the edge of the lake. He can row at a speed of 5 km/h and run at a speed of 7 km/h . The lake has a radius of 3 km with points p,q and r on the circumference of the lake. What is the shortest possible time taken in hours for Owen to row from p to q then run from q to r? (Correct to 2 decimal places)