Answer
The equation of the tangent line is $y=6x-2$.
Work Step by Step
The slope $m$ of the tangent line of the curve $f(x)$ at $A(a,b)$ is also the slope of the curve at that point, which is calculated by $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$f(t)=t^3+3t\hspace{1cm}A(1,4)$$
1) Find the slope $m$ of the tangent:
$$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Here $a=1$ and $f(a)=b=4$
$$m=\lim_{h\to0}\frac{f(h+1)-4}{h}=\lim_{h\to0}\frac{(h+1)^3+3(h+1)-4}{h}$$ $$m=\lim_{h\to0}\frac{h^3+3h^2+3h+1+3h+3-4}{h}$$ $$m=\lim_{h\to0}\frac{h^3+3h^2+6h}{h}=h^2+3h+6$$
$$m=0^2+3\times0+6=6$$
2) Find the equation of the tangent line at $A(1,4)$:
The tangent line would have this form: $$y=6x+m$$
Substitute $A(1,4)$ here to find $m$:
$$6\times1+m=4$$ $$6+m=4$$ $$m=-2$$
So the equation of the tangent line is $y=6x-2$.
