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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 dy⁄dx 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 d⁄dx of both sides of the equation, and then work to isolate dy⁄dx. Once dy⁄dx is isolated, we have the final, simplified form of the differentiated function.
Example Problem
Find the derivative dy⁄dx of the equation cos(y) + x = 5.
Solution:
1.) Taking d⁄dx of both sides of the equation, we get:
d⁄dx[cos(y) + x] = d⁄dx(5)
2.) Differentiating the left side of equation, we get:
d⁄dx[cos(y) + x] = -sin(y)d⁄dx(y) + 1
3.) Differentiating the right side of the equation, we get:
d⁄dx(5) = 0
4.) Simplifying the equation, we get:
-sin(y)d⁄dx(y) + 1 = 0
5.) Isolating the y differential, we get:
d⁄dx(y) = 1/sin(y)
6.) Now that we have dy⁄dx isolated, the solution is:
dy⁄dx = 1/sin(y)