Solved Problems on Implicit Differentiation

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What Are Implicit Functions?

Implicit functions are those that are expressed in terms of both 'x' and 'y', and neither variable is solved in terms of the other. To find the derivative in implicit form, it is not necessary to solve for 'y'. In fact, in some implicit functions it is impossible to solve for 'y'. It is enough to derive member by member, using the rules of differentiation and keeping in mind that:

A. $\text{[math]}$

B. In general $\text{[math]}$

C. For this reason we omit x' and leave y'

D. When the functions are more complex, we use a rule to facilitate the calculation:

$\text{[math]}$

Implicit Function Exercises

Derive the following functions:

Score:

$\text{[math]}$

Solution

1. We derive each term separately. The one containing 'y' with respect to 'y' and the one containing 'x' with respect to 'x'.

$\text{[math]}$

2. We solve for y'

$\text{[math]}$

Solution

1. We derive each term separately. The one containing 'y' with respect to 'y' and the one containing 'x' with respect to 'x'. Terms containing both variables are derived twice, once with respect to 'x' and once with respect to 'y'.

$\text{[math]}$

2. We must solve for y'. We can keep on one side the terms containing y' and move the others to the other side.

$\text{[math]}$

3. We factor out the common factor and solve

$\text{[math]}$

Solution

1. We derive each term separately. In this case we must derive both members, once with respect to 'x' and once with respect to 'y'.

$\text{[math]}$

2. We must solve for y'. We can keep on one side the terms containing y' and move the others to the other side.

$\text{[math]}$

3. We solve the operations with fractions, factor out the common factor, and solve for y'

$\text{[math]}$

Solution

1. We derive each term separately. Those containing both 'x' and 'y' are derived twice, once for each variable. In the second member of the equality we must use the quotient rule for derivatives.

$\text{[math]}$

2. We must solve for y'. We can keep on one side the terms containing y' and move the others to the other side, then solve the operations with fractions.

$\text{[math]}$

Solution

$\text{[math]}$

2. We must solve for y'. We can keep on one side the terms containing y' and move the others to the other side.

$\text{[math]}$

3. We factor out the common factor and solve for y'

$\text{[math]}$

Solution

1. Having several transcendental functions, we move all terms to one side of the equality and apply: $\text{[math]}$

$\text{[math]}$

2. We calculate $\text{[math]}$ and $\text{[math]}$

$\text{[math]}$

3. We substitute into $\text{[math]}$

$\text{[math]}$

Solution

1. Having several transcendental functions, we move all terms to one side of the equality and apply: $\text{[math]}$

$\text{[math]}$

2. We calculate $\text{[math]}$ and $\text{[math]}$

$\text{[math]}$

3. We substitute into $\text{[math]}$

$\text{[math]}$

Solution

1. We calculate $\text{[math]}$ and $\text{[math]}$

$\text{[math]}$

2. We substitute into $\text{[math]}$

$\text{[math]}$

Solution

1. We multiply both members by $\text{[math]}$ to eliminate the fraction and move all terms to one side of the equality

$\text{[math]}$

2. We calculate $\text{[math]}$ and $\text{[math]}$

$\text{[math]}$

3. We substitute into $\text{[math]}$

$\text{[math]}$

Solution

1. Having several transcendental functions, we move all terms to one side of the equality and apply: $\text{[math]}$

$\text{[math]}$

2. We calculate $\text{[math]}$ and $\text{[math]}$

$\text{[math]}$

3. We substitute into $\text{[math]}$

$\text{[math]}$

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Frequently asked questions

🔗 When should you use implicit differentiation instead of just solving for y first?

Implicit differentiation is your best friend when:

Solving for y is impossible or extremely complicated. Some equations can't be solved for y algebraically.
Solving for y gives you a messy expression. Even if you can solve it, the result might be so complex that differentiating it becomes a nightmare.
You only need the derivative at specific points. You don't need y itself, just how it changes.

The beauty is that implicit differentiation gets you the derivative without isolating y, saving time and avoiding algebraic headaches.

📝 What's the key difference in how you treat x and y when deriving implicitly?

When differentiating implicitly:

Treat x as you normally would — its derivative is 1, which you can omit
Treat y differently — remember its derivative is y', not 1
Whenever you differentiate a term with y, multiply by y' (chain rule)
For terms with both x and y (like xy or sin(x + y)), differentiate with respect to both variables separately

This keeps both variables connected, which is the whole point of implicit differentiation.

Chapters

What Are Implicit Functions?
Implicit Function Exercises

Implicit Differentiation Exercises

What Are Implicit Functions?

Implicit Function Exercises

Derivative Calculation Exercises

Derivatives Table