Find the rate of change of the distance d between the origin and a moving point on the graph of y=x^2+7 if dx/dt=2 centimeters per second

Question

Here are the multiple choice answers: A. dD/dt= 4x^3 + 30x/√ x^4 + 15x^2 +49 B. dD/dt= 4x^3 + 16x/x^4 + 14x^2 + 49 C. dD/dt= 2x^3 + 30x/√ x^4 +14x^2 + 49 D. dD/dt= 4x^3 - 30x/√ x^3 +15x^2 + 49 Please show your work and THANK YOU SO MUCH!!!

Answer 1

First, we find a formula for dD/dt.

 

dD/dt = (dD/dx)(dx/dt)

 

Find dD/dx.  This means the derivative of the distance between the points (0,0) and (x, x² + 7).   You need to find the distance using the distance formula.

 

D = √[(0 - x)² + (0 - x² - 7)²]

D = √(x² + (-x² - 7)(-x² - 7)]

D = √(x² + x⁴ + 14x² + 49)

 

D = (x⁴ + 15x² + 49)^1/2

 

 

Take the derivative of this expression using the chain rule, then multiply the result by 2.