Solved: "Find both first partial derivatives. z=ln⁡xy"

College
Calculus and Analysis
Differential Calculus
Derivatives

Efan Halliday

Answered question

2021-02-14

Find both first partial derivatives.

z = \frac{\ln x}{y}

Answer & Explanation

diskusje5

Skilled2021-02-16Added 82 answers

Step 1
Given that
$z = \frac{\ln x}{y}$
Step 2
On partial differentiating z with respect to x and y,
$\frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (\frac{\ln x}{y})$
$= \frac{1}{y} \frac{\partial}{\partial x} (\ln x)$
$= \frac{1}{y} (\frac{1}{x})$
$= \frac{1}{x y}$
And
$\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (\frac{\ln x}{y})$
$= \ln x \frac{\partial}{\partial x} (\frac{1}{y})$
$= \ln x (- \frac{1}{y^{2}})$
$= - \frac{\ln x}{y^{2}}$
Step 3
Hence first partial derivative of $z = \frac{\ln x}{y} i s \frac{\partial z}{\partial x} = \frac{1}{x y} and \frac{\partial z}{\partial y} = - \frac{\ln x}{y^{2}}$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Calculus

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?
How do I find the y-intercept of a parabola?
What are the vertices of $9 x^{2} + 16 y^{2} = 144$ ?
How to determine the rate of change of a function?
Why are the tangents for 90 and 270 degrees undefined?
How to find the center and radius of the circle $x^{2} + y^{2} - 6 x + 8 y = 0$ ?
What is multiplicative inverse of a number?
How to find the continuity of a function on a closed interval?
How do I find the tangent line of a function?
How to find vertical asymptotes using limits?
How to find the center and radius of the circle $x^{2} - 12 x + y^{2} + 4 y + 15 = 0$ ?
Let f be a function so that (below). Which must be true?
I. f is continuous at x=2
II. f is differentiable at x=2
III. The derivative of f is continuous at x=2
(A) I (B) II (C) I and II (D) I and III (E) II and III
How to find the center and radius of the circle given $x^{2} + y^{2} + 8 x - 6 y = 0$ ?
How to find the center and radius of the circle $x^{2} + y^{2} + 4 x - 8 y + 4 = 0$ ?
How to identify the center and radius of the circle ${(x + 3)}^{2} + {(y - 8)}^{2} = 16$ ?

Didn't find what you were looking for?