Implicit derivatives

If y is not a function of x, but we still have a relation between the two through some equation, we can not formulate the derivative as a function of x. This is the case for a circle with radius $a$: $$x^2 + y^2 = a^2$$ A line tangent to this circle will have the slope $ -\frac{y}{x}$. This derivative is implicit, as opposed to explicit as a function of $x$ alone.

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    Sometimes, we run into equations which are completely entangled. Here, the regular way of differentiation falls flat.

    However, with the help of some ingenious tricks, we can still study how the variables relate.

    This is useful for making predictions about price developments in free markets. For instance, in a market economy, the price of any good is determined by the relationship between supply and demand.

    This means that implicit differentiation is a must have for a stock broker!


    Have a look at the unit circle, the circle of radius with the center at the origin, and its equation:

    It's not a function in the form of , since almost every value corresponds to two values.

    However, we don't want to limiting ourselves to only use differentiation on these types of functions. It might still be of interest to perform differentiatiation on this!

    Luckily, we have implicit derivatives to our help.


    Let's return to the equation of the unit circle from the previous slide. Attempting to take the derivative of y directly, we rewrite it as:

    But how do we deal with the symbol? There seem to be at least two derivatives for the same !

    Using implicit derivatives allows us to find a derivative for each pair, with as a function of both and , not only of .

    Continuing with the example of the circle, we find the implicit derivative by taking the derivative of the left and the right side:

    This is the implicit derivative of the unit circle.

    Implicit derivatives

    Implicit functions

    Sometimes you will face equations that, however you twist and turn, it's not possible to disentangle the mess to obtain the form . We call them implicit functions.

    In implicit functions, the is entangled

    An implicit function is actually a function in two variables, and , which can be written as:

    For now, there's no need to dwell upon what a function of two variables is. What we want to do, is differentiation.

    The derivative of the unit circle

    Let's revisit the circle from the introduction to this topic. The mathematical description of the unit circle is:

    In order to take the derivative of this, we try to write it as a function of . Moving the to the other side and taking the square root, we end up with this result:

    Aside from the -sign, this looks all good. It seems we would get at least two derivatives for the same input . The reason for this is that the equation of a circle is not a function in one variable.

    What a bummer. But if we can't get straight to the solution, we might be able to work around it?

    Welcome to the stage: the implicit derivative!

    The trick we will use relies on the chain rule. Let . Note that we do not know how depends on , only that they are related somehow.

    We find the implicit derivative as follows:

    The chain rule comes into play when taking the derivative of .

    Have a look at the final equation. The derivative of depends both on and . Thus, to know the slope of the circle, we need to supply a coordinate pair .

    The derivative of a hyperbola

    Another example of an equation which cause considerable trouble when trying to differentiate it is:

    This is called a hyperbola, and its graph looks like this:

    When attempting to write it as , it doesn't look pretty:

    We go ahead implicitly instead. Applying the same method as for the circle, we get:

    Deriving the power rule

    When talking about differentiation rules, we explored the power rule:

    At that point, we only talked about it for integers . With the implicit derivative in our mind, we shall show that it's valid for , given the power rule with integer exponents.

    Let , with . Which gives, . Taking the derivative of this equation, we obtain:

    We sum it up as a theorem.

    The general power rule

    Applications of the implicit derivative

    The implicit derivative is crucial for being able to solve a type of equations called differential equations. They describe how a quantity relates to its rate of change, and we'll cover them later.

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