# Implicit Differentiation

##### 194 words — categories: calculus

We often write a function the output variable (usually \(y\)) isolated on one side of the equation (e.g., \(y = 3x + 7\)). However, sometimes, it’s cumbersome or impossible to isolate the output variable (e.g. \(y^3 + x^2y^5 - x^4 = 27\)). In such cases, we have what’s called an *implicit* relationship between our variables. We can find the derivative with respect to \(y\) for these functions using a process called **implicit differentiation**. More details about the operation can be found here and here.

In Python, we can perform implicit differentiation using the `sympy.idiff`

.

## Example

For \(y^3 + x^2y^5 - x^4 = 27\) (a) find \(dy/dx\) using implicit differentiation (b) find the slope of the tangent line to the curve at the point \((0, 3)\).

```
from sympy import *
x, y = symbols("x y")
expr = y**3 + x**2*y**5 - x**4 - 27
dydx = idiff(expr, y, x)
dydx
```

`## 2*x*(2*x**2 - y**5)/(y**2*(5*x**2*y**2 + 3))`

`dydx.subs({x:0, y:3})`

`## 0`

## Example

For the demand equation \(x = \sqrt{200 - p^3}\), differentiate implicitly to find \(dp/dx\).

```
p = symbols("p")
expr = -x + sqrt(200 - p**3)
dpdx = idiff(expr, p, x)
dpdx
```

`## -2*sqrt(200 - p**3)/(3*p**2)`