Question

Calculate the derivative using implicit differentiation:

Answer 1

Step-by-step explanation:

Given: $x^5w + w^7 + wz^2 + 8yz = 0$

Taking the partial derivative of the above equation with respect to z, we get

$x^5\dfrac{\partial{w}}{\partial{z}} + 7w^6\dfrac{\partial{w}}{\partial{z}} + z^2\dfrac{\partial{w}}{\partial{z}} + 2z + 8y =0$

Collecting all terms containing $\frac{\partial{w}}{\partial{z}},$ we get

$(x^5 + 7w^6 + z^2)\dfrac{\partial{w}}{\partial{z}} = -2(z + 4y)$

Then

$\dfrac{\partial{w}}{\partial{z}} = \dfrac{-2(z + 4y)}{x^5 + 7w^6 + z^2}$