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.

$\frac{dd}{dt} =$

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.

$\frac{dd}{dt} =$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Related Rates", "Chain Rule Differentiation" ], "new_concepts": [ "Distance Rate of Change" ], "current_concepts": [ "Related Rates", "Chain Rule Differentiation", "Distance Rate of Change" ] } ## Express distance from the origin Using the *Related Rates* knowledge point $$ \begin{aligned} &D = \sqrt{x^2 + y^2}\ &y = x^2\ &D = \sqrt{x^2 + x^4} \end{aligned} $$ ## Differentiate with respect to time Using the *Chain Rule Differentiation* knowledge point $$ \begin{aligned} &\frac{dD}{dt} = \frac{1}{2\sqrt{x^2 + x^4}} \cdot (2x + 4x^3) \cdot \frac{dx}{dt}\ &\frac{dD}{dt} = \frac{x(1 + 2x^2)}{\sqrt{x^2(1 + x^2)}} \cdot \frac{dx}{dt}\ &\frac{dD}{dt} = \frac{x(1 + 2x^2)}{x\sqrt{1 + x^2}} \cdot \frac{dx}{dt} = \frac{1 + 2x^2}{\sqrt{1 + x^2}} \cdot \frac{dx}{dt} \quad (x > 0) \end{aligned} $$ ## Substitute given values Using the *Related Rates* knowledge point $$ \begin{aligned} &x = 9, \quad \frac{dx}{dt} = 4\ &\frac{dD}{dt} = \frac{1 + 2(9)^2}{\sqrt{1 + 9^2}} \cdot 4\ &\frac{dD}{dt} = \frac{1 + 162}{\sqrt{82}} \cdot 4 = \frac{163 \cdot 4}{\sqrt{82}} = \frac{652}{\sqrt{82}}\ &\frac{dD}{dt} = \frac{652\sqrt{82}}{82} = \frac{326\sqrt{82}}{41} \end{aligned} $$ 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. $\frac{dD}{dt} =$ $\frac{326\sqrt{82}}{41}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Related Rates" ] }

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Question

Explanation:

Express distance from the origin

Differentiate with respect to time

Substitute given values

Answer: