Question

Question #7d7dc

Answer 1

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`