find the slope of the functions graph at the given point. then find an equation for the line tangent to the…

find the slope of the functions graph at the given point. then find an equation for the line tangent to the graph there.\n$f(x)=x^{2}+1,(3,10)$\nwhat is the slope of the functions graph at the given point?\n$m = \\square$ (simplify your answer.)

find the slope of the functions graph at the given point. then find an equation for the line tangent to the graph there.\n$f(x)=x^{2}+1,(3,10)$\nwhat is the slope of the functions graph at the given point?\n$m = \\square$ (simplify your answer.)

Answer

Explanation:

Step1: Find the derivative of the function

The function is ( f(x)=x^{2}+1 ). Using the power rule ( (x^{n})^\prime = nx^{n - 1} ), the derivative ( f^\prime(x)=(x^{2}+1)^\prime=(x^{2})^\prime+(1)^\prime ). Since ( (1)^\prime = 0 ) and ( (x^{2})^\prime=2x ), we have ( f^\prime(x)=2x ).

Step2: Evaluate the derivative at ( x = 3 )

To find the slope ( m ) of the tangent line at the point ( (3,10) ), substitute ( x = 3 ) into ( f^\prime(x) ). So ( m=f^\prime(3)=2\times3 ).

Answer:

( 6 )