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Math Geometry Implicit function
2
3
If z
− xz − y = 0,
prove that
2
∂ z
3z
= −
∂x∂y
2
(3z
+ x
3
.
− x)
Question:
2
If z3 − xz − y
prove that
= 0,
2
∂ z
3z
= −
∂x∂y
2
(3z
+ x
.
3
− x)
Implicit Derivative:
Depending on how the function is given implicitly or explicitly, it will be how the partial derivatives of a function of several variables
will be calculated.
For the case of the implicit functions, when calculating the partial derivatives with respect to the whole equation, we will calculate
the derivatives with respect to one of the variables, considering the rest of the independent variables as constants.
Answer and Explanation: 1
First, we calculate the partial derivative with respect to x of the whole equation:
3
∂z
2
z
∂z
− xz − y = 03z
∂x
2
∂z
∂z
− 1 = 0
∂x
2
3z
− x
1
=
∂y
z
=
:
∂z
− x
∂y
= 0
∂x
The same for the variable y
3z
∂z
− z − x
∂y
2
3z
− x
Calculating the derivative with respect to x of the partial derivative with respect to y
∂z
∂z
1
=
∂y
2
3z
0 − 1 (6z
2
=
− x ∂x∂y
2
(3z
z
−6z (
2
) + 1
2
3z
− x
=
2
∂x∂y
(3z2 − x)
2
2
−6z
+ (3z
2
3z
=
2
(3z
2
2
− x)
2
−6z
+ 3z
2
(3z
2
2
∂ z
3z
= −
∂x∂y
2
(3z
− x)
− x
− x
=
2
− x)
Simplifying, we check the result:
∂ z
∂z
− 1)
−6z
∂x
∂ z
3
− x)
+ x
3
− x)
Help improve Study.com. Report an Error
+ 1
∂x
=
2
(3z
2
− x)
:
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Learn more about this topic:
Implicit Functions
from
Chapter 1 / Lesson 11
11K
Discover the implicit function and learn how it is used for equations. Explore how implicit equations work and study some examples
of implicit functions.
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