Find the rate of change of the distance between the origin and a moving point on the graph of $y = x ^ { 2 } + 7$ if $\frac { d x } { d t } = 6$ centimeters per second.
A) $\frac { d D } { d t } = \frac { 12 x ^ { 3 } + 90 x } { \sqrt { x ^ { 4 } + 15 x ^ { 2 } + 49 } }$
B) $\frac { d D } { d t } = \frac { 6 x ^ { 3 } - 20 x } { x ^ { 3 } + 15 x ^ { 2 } + 49 }$
C) $\frac { d D } { d t } = \frac { 12 x ^ { 3 } - 90 x } { \sqrt { x ^ { 3 } + 15 x ^ { 2 } + 49 } }$
D)$\frac { d D } { d t } = \frac { 6 x ^ { 3 } + 90 x } { \sqrt { x ^ { 4 } + 14 x ^ { 2 } + 49 } }$
E)$\frac { d D } { d t } = \frac { 12 x ^ { 3 } + 20 x } { x ^ { 4 } + 12 x ^ { 2 } + 49 }$

Question

Quizplus · Accepted Answer

The distance $D$ from the origin to a point $(x, y)$ on the graph is given by $D = \sqrt{x^2 + y^2}$. Substituting $y = x^2 + 7$ gives $D = \sqrt{x^2 + (x^2 + 7)^2}$. Differentiating $D$ with respect to $t$ using the chain rule and given $\frac{dx}{dt} = 6$, we find the rate of change of $D$ with respect to $t$. The correct differentiation leads to the formula in option A.

Find the Rate of Change of the Distance Between the Origin