Differentiable Function Problem The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions: $f(0)=-2$, $f'(0)=3$, $f''(0)=-1$.?
A) The function $g$ is given by $g(x)=\tan(ax)+f(x)$ for all real numbers, where $a$ is a constant. Find $g'(0)$ and $g''(0)$ in terms of $0$.
B) The function $h$ is given by $h(x)=\sin(kx) \cdot f(x)$ for all real numbers, where $k$ is a constant. Find $h'(x)$ and write an equation for the line tangent to the graph of $h$ at $x=0$.
I'm confused on how exactly to start the problem properly and how the conditions $(f(0)=-2, f'(0)=3, f''(0)=-1)$ are used when finding $g'(0)$ and $g''(0)$.

A) 
\begin{align}
g(x)=\tan(ax)+f(x) \Longrightarrow g'(x)&=a\sec^2(ax)+f'(x) \\ g''(x)&=2a^2\sec^2(ax)\tan(ax)+f''(x),
\end{align} in particular 
\begin{align}
g'(0) &=a\sec^2(a\cdot 0)+f'(0)=2a^2\sec(0)+f'(0)\\
      &=a\cdot 1+(-2)\\
      &=a-2 \\
g''(0)&=2a^2\sec^2(0)\tan(0)+f''(0)=2a^2(1)(0)+f''(0)\\
      &=0+(-1)\\
      &=-1
\end{align}
Part B) can be solved in a similar form.
