How to Differentiate Implicit Functions with Formula and Solved Examples
Differentiation is one of the building blocks of calculus. It calculates the rate of change of a given quantity. Today, we use differentiation in almost every aspect of physics, mathematics, and chemistry. Do you want to calculate the acceleration of a sprinter exactly after 5 seconds of start? Or do you want to calculate the displacement of a moving body traveling at a certain speed in the given time? You can apply differential calculus in various real-time problems and find the correct solution.
Differential calculus is the area of study that calculates the derivatives of a function, and the process of calculating derivatives is known as differentiation. For a given function y=f(x), the derivative of y is (dy/dx) where dy represents the change in y and dx represents the change in x.
(tangent on a function y=f(x))
In the above graph, we can calculate the derivative of the given function y=f(x)by determining the slope of the tangent line P. We mainly use differentiation on two types of functions: explicit function and implicit function. So, you must be wondering what is the Implicit Function?
Implicit Function
Implicit Function definition conveys when we are not able to isolate the dependent variable in an equation that function becomes an implicit function. Both the dependent variables and independent variables are present in this type of function. For eg: x2 +y2= 16 . Whereas explicit function is present in terms of the independent variable. Consider y=f(x) where x is an independent variable, and y is a dependent variable.
In explicit function, y is in terms of x. According to implicit function meaning, a function is expressed in both y and x.
Differentiation Of Implicit Functions
We can easily differentiate explicit functions using chain rule or rule of differentiation. But sometimes, it can be challenging to express an equation explicitly. Suppose if we were to reduce the equation x2 +y2= 16 in terms of x them we get
Y = + \[\sqrt{(16 - x ^{2}}\] and y = - \[\sqrt{(16 - x ^{2}}\] . We perform implicit differentiation in such cases.
The implicit differentiation meaning isn’t exactly different from normal differentiation. Since we cannot reduce implicit functions explicitly in terms of independent variables, we will modify the chain rule to perform differentiation without rearranging the equation. In the above example, we will differentiate each term in turn, so the derivative of y2 will be 2y*dy/dx. Let us look at implicit differentiation examples to understand the concept better.
Solved Examples
Example 1:
What is implicit function differentiation of x2 +y2= 16?
Answer:
Using implicit function definition
We will perform Differentiation of implicit functions on both sides and each term w.r.t x.
2x + 2y(dy/dx)=0
Rearranging the equation
2y(dy/dx)=-2x
Solving the equation
dy/dx= -2x/2y
Derivative of implicit function is dy/dx= -x/y
Let us look at some other examples.
Example 2:
Find dy/dx If y=sin(x) + cos(y)
Answer:
According to implicit function meaning the given function is implicit.
Hence, we will calculate the derivative of implicit function without rearranging the equation.
Performing Differentiation of implicit functions on both sides and each terms with respect to x.
dy/dx=cos(x)-sin(y)*dy/dx
Rearranging the above equation
dy/dx+sin(y)*dy/dx=cos(x)
dy/dx(1+sin(y))=cos(x)
Solve the equation
dy/dx=cos(x)/1+sin(y)
Example 3:
Differentiate 10x4 - 18xy2 + 10y3 = 48 with respect to x.
Answer:
The implicit function meaning holds true for the given function.
Hence, we will use the product rule of differentiation on xy2 i.e (FG)’ = F G’ + F’ G
Let us find out the derivative of Implicit function by differentiating each term in the equation w.r.t x.
10 (4x2) - 18(x(2y * dydx) + y2) + 10(3y2 * dydx) = 0
Let us further simplify the above equation.
40x3 - 36xy * dydx - 18y2 + 30y2 * dydx = 0
Now we bring all dy/dx on the left side and rest of the terms on the right side
-36xy * dydx + 30y2 * dydx = 40x3 + 18y2
Taking dy/dx common
(30y2 - 36xy) dydx = 18y2 - 40x2
Divide both sides by two and solve the equation
dy/dx = 9y2 - 20x2/(15y2 - 18xy)
FAQs on Implicit Function Differentiation Explained with Concept and Applications
1. What is implicit function differentiation?
Implicit function differentiation is the process of finding dy/dx when y is defined implicitly in terms of x, rather than explicitly as y = f(x). In implicit differentiation, both variables are treated as functions of x and differentiated accordingly.
- Used when equations are in the form F(x, y) = 0.
- Apply differentiation to every term with respect to x.
- Use the chain rule when differentiating terms containing y.
2. How do you do implicit differentiation step by step?
To perform implicit differentiation, differentiate both sides of the equation with respect to x and solve for dy/dx. Follow these steps:
- Differentiate every term with respect to x.
- When differentiating terms with y, multiply by dy/dx (chain rule).
- Collect all dy/dx terms on one side.
- Factor out dy/dx and solve.
- 2x + 2y(dy/dx) = 0
- 2y(dy/dx) = −2x
- dy/dx = −x/y
3. What is the formula for implicit differentiation?
The general formula for implicit differentiation from F(x, y) = 0 is dy/dx = −(∂F/∂x)/(∂F/∂y), provided ∂F/∂y ≠ 0. This formula comes from differentiating both sides with respect to x and solving for dy/dx.
- Useful for complex implicit functions.
- Requires partial derivatives.
- Applies when y is differentiable with respect to x.
4. Why do we use the chain rule in implicit differentiation?
We use the chain rule because y is treated as a function of x when differentiating implicitly. Whenever you differentiate a term involving y, you must multiply by dy/dx.
- Example: d/dx (y²) = 2y(dy/dx).
- This accounts for the hidden dependence of y on x.
- Without the chain rule, the derivative would be incomplete.
5. Can you give an example of implicit differentiation?
Yes, for the equation xy + y² = 10, implicit differentiation gives dy/dx = −y/(x + 2y). Solution steps:
- Differentiate: x(dy/dx) + y + 2y(dy/dx) = 0.
- Group dy/dx terms: (x + 2y)(dy/dx) + y = 0.
- Solve: (x + 2y)(dy/dx) = −y.
- Final answer: dy/dx = −y/(x + 2y).
6. What is the difference between implicit and explicit differentiation?
The main difference is that explicit differentiation works on functions written as y = f(x), while implicit differentiation works on equations where y is not isolated. Key differences:
- Explicit: Directly differentiate f(x).
- Implicit: Differentiate both sides of F(x, y) = 0.
- Implicit requires the chain rule for y terms.
7. How do you find the second derivative using implicit differentiation?
To find the second derivative d²y/dx², differentiate the first derivative implicitly again with respect to x. Steps:
- First find dy/dx using implicit differentiation.
- Differentiate dy/dx again with respect to x.
- Apply the chain rule where needed.
- Simplify the result.
8. When can you use implicit differentiation?
You can use implicit differentiation when x and y are related by an equation that cannot be easily rearranged into y = f(x). It is especially useful for:
- Circles (x² + y² = r²).
- Ellipses and conic sections.
- Equations involving both x and y mixed together.
9. What are common mistakes in implicit differentiation?
A common mistake in implicit differentiation is forgetting to multiply by dy/dx when differentiating terms containing y. Other frequent errors include:
- Not applying the chain rule correctly.
- Forgetting product rule in terms like xy.
- Failing to solve completely for dy/dx.
10. How is implicit differentiation used in real life?
Implicit differentiation is used in real-life applications involving related rates and multivariable relationships. Common uses include:
- Related rates problems in physics and engineering.
- Economics models with dependent variables.
- Geometry problems involving curves.
