Use the limit definition to find the slope of the tangent line to the graph of $f ( x ) = \sqrt { 4 x + 25 }$ at the point $( 6,7 )$ .
A) $\frac { 2 } { 7 }$
B) $- \frac { 2 } { 7 }$
C) $\frac { 1 } { 7 }$
D) $- \frac { 1 } { 7 }$
E) $\frac { 1 } { 6 }$

Question

Quizplus · Accepted Answer

The slope of the tangent line to the graph of $f(x) = \sqrt{4x + 25}$ at a point can be found using the derivative of $f(x)$ with respect to $x$, which gives the slope of the tangent line at any point $x$. The derivative of $f(x) = \sqrt{4x + 25}$ using the chain rule is $\frac{1}{2\sqrt{4x + 25}} \cdot 4 = \frac{2}{\sqrt{4x + 25}}$. Substituting $x = 6$ into this derivative gives $\frac{2}{\sqrt{4(6) + 25}} = \frac{2}{\sqrt{49}} = \frac{2}{7}$, which is the slope of the tangent line at the point $(6,7)$.

Use the Limit Definition to Find the Slope of the Tangent