find $\frac{dx}{dt}$ at $x = -5$ if $y = 5x^2 - 4$ and $\frac{dy}{dt} = 4$.

$\frac{dx}{dt} =$

Question

find $\frac{dx}{dt}$ at $x = -5$ if $y = 5x^2 - 4$ and $\frac{dy}{dt} = 4$.

$\frac{dx}{dt} =$

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" ], "new_concepts": [], "current_concepts": [ "Related Rates", "Implicit Differentiation" ] } ## Differentiate the equation with respect to time Using the *Related Rates* knowledge point $$ \begin{aligned} &y = 5x^2 - 4\ &\frac{d}{dt}(y) = \frac{d}{dt}(5x^2 - 4)\ &\frac{dy}{dt} = 10x \frac{dx}{dt} \end{aligned} $$ ## Substitute the given values and solve Using the *Related Rates* knowledge point $$ \begin{aligned} &4 = 10(-5) \frac{dx}{dt}\ &4 = -50 \frac{dx}{dt}\ &\frac{dx}{dt} = -\frac{4}{50} = -\frac{2}{25} \end{aligned} $$ Find $\frac{dx}{dt}$ at $x = -5$ if $y = 5x^2 - 4$ and $\frac{dy}{dt} = 4$. $\frac{dx}{dt} =$ $-\frac{2}{25}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Related Rates" ] }

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Question

Explanation:

Differentiate the equation with respect to time

Substitute the given values and solve

Answer: