Step by step solution to "Use implicit differentiation to find ∂z / ∂x..."

Yolanda Jorge

Answered question

2021-11-11

Use implicit differentiation to find ∂z / ∂x and ∂z / ∂y.

e^{4 z} = x y z

Louise Eldridge

Beginner2021-11-12Added 17 answers

Given:

\Rightarrow e^{4 z} = x y z

Partial differentiating both the sides with respect to x:

\Rightarrow \frac{d}{d x} (e^{4 z}) = \frac{d}{d x} (x y z)

\Rightarrow e^{4 z} \frac{d z}{d x} = x y \frac{d z}{d x} + y z

\Rightarrow e^{4 z} \frac{d z}{d x} - x y \frac{d z}{d x} = y z

\Rightarrow \frac{d z}{d x} (e^{4 z} - x y) = y z

\Rightarrow \frac{d z}{d x} = \frac{y z}{e^{4 x} - x y}

Partial differentiating both sides with respect to y:

\frac{d}{d y} (e^{4 z}) = \frac{d}{d y} (x y z)

\Rightarrow e^{4 z} \frac{d z}{d y} = x y \frac{d z}{d y} + x z

\Rightarrow e^{4 z} \frac{d z}{d y} - x y \frac{d z}{d y} = x z

\Rightarrow \frac{d z}{d y} (e^{4 z} - x y) = x z

\Rightarrow \frac{d z}{d y} = \frac{x z}{e^{4 z} - x y}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?