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Implicit Differentiation Calculator

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Implicit Differentiation Lesson

Lesson Contents

How to Perform Implicit Differentiation

The calculator above finds the value of your derivative order input by using the process known as implicit differentiation. It uses similar steps to standard paper and pencil Calculus, but much faster than what a human being is capable of.

Using implicit differentiation to calculate a derivative is useful when the dependent variable is not isolated on one side of the equation (usually y is the dependent variable). When the dependent variable is on the same side of the equation as the independent variable and cannot be simply subtracted or added to the other side to isolate it, implicit differentiation is necessary.

An equation like such is called an implicit relation because one of the variables is an implicit function of the other. An example of an implicit relation is sin(xy) = 2. If we wanted to calculate the derivative dydx of this equation, we are unable to use the usual trigonometry derivative rules to differentiate the sin term with x because that term also has y in it.

The general process for implicit differentiation is to take the derivative of both sides of the equation, and then isolate the full differential operator. For example, in the case where the equation has y as the dependent variable and x as the independent variable, we would take ddx of both sides of the equation, and then work to isolate dydx. Once dydx is isolated, we have the final, simplified form of the differentiated function.

Example Problem

Find the derivative dydx of the equation cos(y) + x = 5.

1.) Taking ddx of both sides of the equation, we get:
ddx[cos(y) + x] = ddx(5)

2.) Differentiating the left side of equation, we get:
ddx[cos(y) + x] = -sin(y)ddx(y) + 1

3.) Differentiating the right side of the equation, we get:
ddx(5) = 0

4.) Simplifying the equation, we get:
-sin(y)ddx(y) + 1 = 0

5.) Isolating the y differential, we get:
ddx(y) = 1/sin(y)

6.) Now that we have dydx isolated, the solution is:
dydx = 1/sin(y)

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