# What is the implicit derivative of #5=x-1/(xy^2)#?

##### 2 Answers

See below.

#### Explanation:

When finding the implicit derivative of an equation, we are finding the derivative in respect to

This means that when we find the derivative of any variable other than

Knowing this, we will start with implicit differentiation.

Here is what we are given:

We can go ahead and avoid the Quotient Rule by multiply each term by

Here we can use the Product Rule to solve:

Simplify:

Solve for

Please see below.

#### Explanation:

I assume that we want

**Method 1**

To avoid the quotient rule, I would start by multiplying both sides be

Differentiating with respect to

Now we'll do some algebra.

# = ( 2xy - 5y)/(10x - 2x^2)#

**Method 2**

**Resolution**

To see that these are equivalent, start with

# = ( 2xy - (5)y)/(2x(5) - 2x^2)#

Now replace

# = ( 2xy - (x-1/(xy^2))y)/(2x(x-1/(xy^2)) - 2x^2)#

Simplify algebraically to get

# = - (x^2y^3+y)/(2x)#