Related rates from an equation practice problems
Differentiate an equation with respect to time and solve for the unknown rate. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x y = 2 \). If \( \displaystyle \frac{dx}{dt} = -4 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 1 \) and \( \displaystyle y = 2 \).
If \( \displaystyle y = x^{2} + 3 \) and \( \displaystyle \frac{dx}{dt} = 3 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = -1 \).
If \( \displaystyle y = x^{3} - 2 x \) and \( \displaystyle \frac{dx}{dt} = 4 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 2 \).
If \( \displaystyle y = 4 x^{2} + x \) and \( \displaystyle \frac{dx}{dt} = 4 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = -3 \).
If \( \displaystyle y = x^{3} - x \) and \( \displaystyle \frac{dx}{dt} = -4 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 2 \).
\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x^{2} + y^{2} = 100 \). If \( \displaystyle \frac{dx}{dt} = -4 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 6 \) and \( \displaystyle y = 8 \).
If \( \displaystyle y = x^{3} - 2 x \) and \( \displaystyle \frac{dx}{dt} = -3 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 3 \).
If \( \displaystyle y = x^{2} - 4 \) and \( \displaystyle \frac{dx}{dt} = -5 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 3 \).
\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x^{2} + y^{2} = 169 \). If \( \displaystyle \frac{dx}{dt} = -2 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 12 \) and \( \displaystyle y = 5 \).
If \( \displaystyle y = \sqrt{x + 2} \) and \( \displaystyle \frac{dx}{dt} = 2 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 3 \).