For the function f(x) = \frac{1}{\sqrt x} , find an equation of the tangent line the graph of...

Question:

For the function {eq}f(x) = \frac{1}{\sqrt x} {/eq}, find an equation of the tangent line the graph of the above function at {eq}(4, \frac{1}{2}) {/eq}

Tangent Line:

The tangent line determines a first degree approximation of the function, for differentiable functions, we have the general expression of the tangent line as: {eq}y = f(x),x = a \to y = f(a) + f'(a)\left( {x - a} \right). {/eq}

Answer and Explanation:

Taking into account the function and the given value:

{eq}f(x) = \frac{1}{{\sqrt x }}\quad ,\left( {4,\frac{1}{2}} \right) \to f(4) = \frac{1}{2} {/eq}

With the first derivative, at the point:

{eq}f(x) = \frac{1}{{\sqrt x }} = {x^{ - 1/2}}\\ f'(x) = - \frac{1}{2}{x^{ - 3/2}}\\ f'(4) = - \frac{1}{2}{4^{ - 3/2}} = - \frac{1}{{16}} {/eq}

We can write the tangent line as:

{eq}y = f(4) + f'(4)\left( {x - 4} \right)\\ y = \frac{1}{2} - \frac{1}{{16}}\left( {x - 4} \right) {/eq}


Learn more about this topic:

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Tangent Line: Definition & Equation

from NY Regents Exam - Geometry: Tutoring Solution

Chapter 1 / Lesson 11
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