a. Use the definition $m_{\tan }=\underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}$ to find the slope of the line tangent to the graph of $f$ at $P$.
b. Determine an equation of the tangent line at $P$.
$$f(x)=x^4,P(2,16)$$
a. $m_{\tan }=32$
b. $y=$

Step 1 ：Given the function $f(x)=x^4$ and the point $P(2,16)$.

Step 2 ：The slope of the tangent line to the graph of the function at a point is given by the derivative of the function at that point.

Step 3 ：Find the derivative of the function $f(x)=x^4$, which is $f^{\prime }(x)=4x^3$.

Step 4 ：Evaluate the derivative at $x=2$ to find the slope of the tangent line, which is $f^{\prime }(2)=4\ast 2^3=32$.

Step 5 ：Use the point-slope form of the equation of a line, $y-y_1=m(x-x_1)$, to find the equation of the tangent line. Here, $m=32$ and $(x_1,y_1)=(2,16)$.

Step 6 ：Substitute the values into the equation to get $y-16=32(x-2)$.

Step 7 ：Simplify the equation to get $y=32x-48$.

Step 8 ：Final Answer: The equation of the tangent line to the graph of the function $f(x)=x^4$ at the point $P(2,16)$ is $\boxed{{\displaystyle y=32x-48}}$.