The graphs of $f$ and $g$ are shown. Find $K^{\prime }\left(6\right)$ if $K=5f-2g$.

Consider the curve defined by $y=x^3-8x+5$. Determine the smallest slope of the curve and the point at which it occurs.

Determine the first and second derivatives of $y=x^3+x^2-10$.

Find the derivative of the function $g\left(x\right)=e^x(5x^3-15x^2+30x-30)$.

Calculate the derivative of the function $y=\sqrt{x}\sin \left(\right(4x{)}^4)$.

Apply L'Hôpital's rule to find the limit of the following function as $x\rightarrow0$:

$g\left(x\right)=\frac{\frac{1}{2{({(1+x)}^6+4)}^9}-\frac{1}{2{\left(5\right)}^9}}{x}$

Find the derivative of $y=\frac{1}{8}{(2+{\cos }^3\left(6t\right))}^4$.

Find the maximum slope of the tangent line to the curve $y=5\sin \left(\frac{3\pi x}{2}\right)$ within the interval .