Implicit Differentiation
Implicit differentiation is an approach to taking derivatives that uses the chain rule to avoid solving explicitly for one of the variables. For example, if we can directly take the derivative of each term with respect to to obtain so
Implicit differentiation is especially useful where it is difficult to isolate one of the variables in the given relationship. For example, if solving for and then taking the derivative would be painful. Instead, using implicit differentiation to directly take the derivative with respect to gives
so This can be especially useful in fields like physics where relationships between variables are often known but a nice closed-form is not attainable.
Contents
- Implicit Differentiation - Polynomials
- Implicit Differentiation - Radical Functions
- Implicit Differentiation - Rational functions
- Implicit Differentiation - Trigonometric Functions
- Implicit Differentiation - Inverse Trigonometric Functions
- Implicit Differentiation - Exponential and Logarithmic Functions
- Implicit Differentiation - Related Rates
- Problem Solving
Implicit Differentiation - Polynomials
Given , what is at the point
Since , differentiating, we have
Thus at the point is .
Alternatively, we can use the fact that if , then Rewriting the function,
Applying the method given above,
Thus at the point is .
Implicit Differentiation - Radical Functions
Implicit Differentiation - Rational functions
Implicit differentiation is a technique that can be used to differentiate equations that are not given in the form of For instance, the differentiation of looks pretty tough to do by using the differentiation techniques we've learned so far (which were explicit differentiation techniques), since it is not given in the form of Observe that the term equations was used in the first sentence. In fact, implicit differentiation can be used to differentiate equations that do not match the criteria to be a function.
The main key to implicit differentiation is to keep in mind with respect to which variable we are differentiating. Here is an easy explanation of how it is done:
When differentiating an equation implicitly with respect to we should
(1) add to all of the terms;
(2) apply the iterated chain rule to any terms that are not expressed in terms of
The first step is the process of differentiating both sides of an equation with respect to The second step gives the solution to terms that are not described by Remember the iterated chain rule? Here is a quick reminder:
Implicit differentiation can be used in differentiating rational functions for convenience. Although most rational functions are given in the form of differentiating these explicitly would require using the quotient rule, which is pretty annoying. A slight modification to the equation (which in most cases would be multiplying the quotient to both sides) and the application of implicit differentiation would ease up obtaining the derivative. Here are some examples:
Find the derivative of using implicit differentiation.
To avoid using the quotient rule, we can change both sides to their reciprocals:
Then we differentiate both sides:
and apply the chain rule:
Find the derivative of using implicit differentiation.
First, we multiply both sides by to get rid of the quotient:
Then we differentiate both sides of the equation. Obviously, the product rule should be used for the left-hand side:
If you don't like having the term in the answer, you can plug in which gives
However, this procedure isn't necessary since we can always find the value of when given the value of
Of course, using explicit differentiation by applying the quotient rule will also lead to the same answer. Try it out for yourself.
Find the derivative of using implicit differentiation.
We have
Find the tangent to the curve at
First, we should find the slope of the tangent, which is equal to We have
Since we have
Therefore, the equation of the tangent is
Find the derivative of
First, we add to both sides of the equation, which gives
Then we use the chain rule to solve
Notice that the answer can contain variables other than for it would be messy to express the equation using only one variable.
Find the derivative of
We have
Find the derivative of
We have
Find the tangent to the curve at
First, we find the slope which is equal to Differentiating both sides with respect to gives
Therefore the slope of the tangent is
Since the tangent passes through the point the equation of the tangent is
Implicit Differentiation - Trigonometric Functions
Find if
Using implicit differentiation,
Implicit Differentiation - Inverse Trigonometric Functions
We can also use implicit differentiation to find the derivatives of the inverse trigonometric functions, assuming that these functions are differentiable.
Find the derivative of using implicit differentiation.
Since we have where
Differentiating implicitly, we have
Since and using
we have
Implicit Differentiation - Exponential and Logarithmic Functions
Implicit differentiation is an approach to taking derivatives that uses the chain rule to avoid solving explicitly for one of the variables.
If what is
Taking the derivative of every term with respect to gives us
Now, we can isolate all of the on the left:
Problem Solving