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$f(x)=\\sqrt{7 x+1},(5,6)$\nthe slope of the functions graph at $(5,6)$ is □.\n(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$f(x)=\\sqrt{7 x+1},(5,6)$\nthe slope of the functions graph at $(5,6)$ is □.\n(simplify your answer.)

Answer

Explanation:

Step1: Use the power rule for differentiation

The function is ( f(x)=\sqrt{7x + 1}=(7x + 1)^{\frac{1}{2}} ). By the chain - rule ( (u^n)^\prime=nu^{n - 1}\cdot u^\prime ), where ( u = 7x+1), (n=\frac{1}{2}). First, (u^\prime=7). Then (f^\prime(x)=\frac{1}{2}(7x + 1)^{\frac{1}{2}-1}\cdot7=\frac{7}{2}(7x + 1)^{-\frac{1}{2}}=\frac{7}{2\sqrt{7x + 1}}).

Step2: Evaluate the derivative at (x = 5)

Substitute (x = 5) into (f^\prime(x)). When (x = 5), (f^\prime(5)=\frac{7}{2\sqrt{7\times5+1}}=\frac{7}{2\sqrt{35 + 1}}=\frac{7}{2\sqrt{36}}=\frac{7}{2\times6}=\frac{7}{12}).

Answer:

(\frac{7}{12})