Question

find the rate of change of the distance \\(d\\) from the origin of a point moving on the graph of \\(f(x) = x^2\\) when \\(x = 9\\) and \\(\frac{dx}{dt} = 4\\) units per second. write the exact answer. do not round.

answer
\\(\frac{dd}{dt} =\\)

Explanation:

Express distance from the origin

Using the Related Rates knowledge point

$$ D = \sqrt{x^2 + y^2} $$

Since the point lies on the curve \(y = x^2\), we substitute \(y\) to express \(D\) solely in terms of \(x\):

$$ D = \sqrt{x^2 + (x^2)^2} = \sqrt{x^2 + x^4} $$

Differentiate with respect to time

Using the Chain Rule Differentiation knowledge point

$$ \frac{dD}{dt} = \frac{1}{2\sqrt{x^2 + x^4}} \cdot (2x + 4x^3) \cdot \frac{dx}{dt} = \frac{x + 2x^3}{\sqrt{x^2 + x^4}} \cdot \frac{dx}{dt} $$

Substitute the given values

Using the Related Rates knowledge point

$$
LATEXBLOCK0
$$

Simplify the exact expression

Using the Related Rates knowledge point

$$
LATEXBLOCK1
$$

Answer:

Find the rate of change of the distance \(D\) from the origin of a point moving on the graph of \(f(x) = x^2\) when \(x = 9\) and \(\frac{dx}{dt} = 4\) units per second. Write the exact answer. Do not round.

\(\frac{dD}{dt} =\) <blank>\(\frac{326\sqrt{82}}{41}\)</blank>