Calculus Problem

Question

let xy=4 and (dy/dt)=1 . find (dx/dt) when x=2

f(x)= ?

Answer 1

We can use implicit differentiation and the product rule to solve. Thus:

 

D_t(xy) = D_t(4) [take derivative of both sides of the equation with respect to t]

(dx/dt)*y + x*(dy/dt) = 0 [product rule]

(dx/dt)*y + 2*1 = 0 [substitution with x=2 and (dy/dt)=1 as given]

(dx/dt)*y = -2 [algebra]

 

Now if x=2, then by the original equation, xy=4, y must also be 2 because 2*2 = 4, so we get:

 

(dx/dt)*2 = -2 [substitution y=2]

(dx/dt) = -1 [algebra]