Question #7d7dc

1 Answer
Dec 7, 2017

You need to use implicit differentiation. Please see the explanation.

Explanation:

Given:

#PV^(7/5)=c#

Differentiate with respect to P:

#(d(PV^(7/5)))/(dP)=(d(c))/(dP)#

The term on the left is a product of two variables, therefore, we use the product rule:

#(d(P))/(dP)V^(7/5)+ P(d(V^(7/5)))/(dP) = (d(c))/(dP)#

The derivative of P with respect to P is 1:

#V^(7/5)+ P(d(V^(7/5)))/(dP) = (d(c))/(dP)#

We use the chain rule on the derivative in the second term:

#V^(7/5)+ P(d(V^(7/5)))/(dV)(dV)/(dP) = (d(c))/(dP)#

Because the derivative is with respect to V, we may apply the power rule:

#V^(7/5)+ 7/5PV^(2/5)(dV)/(dP) = (d(c))/(dP)#

The derivative of the constant on the right is 0:

#V^(7/5)+ 7/5PV^(2/5)(dV)/(dP) = 0#

Solve the above equation for #(dV)/(dP)#:

#7/5PV^(2/5)(dV)/(dP) = -V^(7/5)#

#(dV)/(dP) = -V^(7/5)/(7/5PV^(2/5))#

#(dV)/(dP) = -5/7V/P#