Answer
The equation of the tangent line is $y=4x-3$.
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 $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$y=f(x)=x^2+1\hspace{1cm}A(2,5)$$
1) Find the slope $m$ of the tangent:
$$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Here $a=2$ and $f(a)=b=5$
$$m=\lim_{h\to0}\frac{f(2+h)-5}{h}=\lim_{h\to0}\frac{(2+h)^2+1-5}{h}$$ $$m=\lim_{h\to0}\frac{4+4h+h^2+1-5}{h}=\lim_{h\to0}\frac{4h+h^2}{h}$$ $$m=\lim_{h\to0}(4+h)$$ $$m=4+0=4$$
2) Find the equation of the tangent line at $A(2,5)$:
The tangent line would have this form: $$y=4x+m$$
Substitute $A(2,5)$ here to find $m$:
$$4\times2+m=5$$ $$8+m=5$$ $$m=-3$$
So the equation of the tangent line is $y=4x-3$.
