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.\nf(x)=x² + 1, (-5,26)\nwhat is the slope of the functions graph at the given point?\nm = 27 (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.\nf(x)=x² + 1, (-5,26)\nwhat is the slope of the functions graph at the given point?\nm = 27 (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)=2x).

Step2: Calculate the slope at the given point

The (x) - coordinate of the point ((-5,26)) is (x=-5). Substitute (x = - 5) into (f^\prime(x)): (m=f^\prime(-5)=2\times(-5)=-10).

Step3: Use the point - slope form to find the tangent line equation

The point - slope form is (y - y_1=m(x - x_1)), where ((x_1,y_1)=(-5,26)) and (m=-10). Substitute the values: (y - 26=-10(x + 5)). Expand: (y-26=-10x-50). Add 26 to both sides: (y=-10x-24).

Answer:

The slope (m = - 10) and the equation of the tangent line is (y=-10x - 24)