Solving Calculus Problems On Implicit Differentiation

 

Understanding Implicit Differentiation

In calculus, we usually work with explicit functions, where one variable is isolated directly on one side of the equation—for example, y = x² + 3x. Differentiating these is straightforward because y is stated explicitly in terms of x.

However, many equations mix x and y together in a way that is difficult or impossible to solve directly for y. These are called implicit functions (such as x² + y² = 25 or x³ + y³ = 3xy).

To find ^dy⁄_dx for these expressions, we use Implicit Differentiation. The process follows three core steps:

1. Differentiate both sides of the equation with respect to x.
2. Treat y as a function of x. This means every time you differentiate a term containing y, you must apply the Chain Rule and multiply that term by ^dy⁄_dx.
3. Use algebra to isolate and solve for ^dy⁄_dx.

Guided Example

Problem: Find ^dy⁄_dx for the curve x² + y² = 36.

Step 1: Differentiate term by term with respect to x:
The derivative of x² is 2x.
The derivative of y² (using the Chain Rule) is 2y · ^dy⁄_dx.
The derivative of the constant 36 is 0.

Putting it together:
2x + 2y(^dy⁄_dx) = 0

Step 2: Isolate the ^dy⁄_dx term:
Subtract 2x from both sides:
2y(^dy⁄_dx) = -2x

Divide both sides by 2y:
^dy⁄_dx = ^-2x⁄_2y

Simplifying gives the final result:
^dy⁄_dx = -^x⁄_y

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Implicit Differentiation Practice Quiz

Question 1: Find ^dy⁄_dx if y³ - x³ = 8.

• A) ^x2⁄_y²
• B) -^x2⁄_y²
• C) ^y2⁄_x²
• D) 3x²

View Answer & Solution

Correct Answer: A

Solution:
1. Differentiate both sides with respect to x:
3y²(^dy⁄_dx) - 3x² = 0
2. Move the x term to the right side:
3y²(^dy⁄_dx) = 3x²
3. Divide by 3y² to isolate the derivative:
^dy⁄_dx = ^3x2⁄_3y² = ^x2⁄_y²

Question 2: Find the derivative ^dy⁄_dx for the curve expression xy = 4.

• A) ¹⁄_x
• B) -^y⁄_x
• C) -^x⁄_y
• D) 0

View Answer & Solution

Correct Answer: B

Solution:
1. Use the Product Rule on the term xy: (first · derivative of second) + (second · derivative of first).
x(^dy⁄_dx) + y(1) = 0
2. Isolate the derivative term:
x(^dy⁄_dx) = -y
3. Divide by x:
^dy⁄_dx = -^y⁄_x

Question 3: Differentiate implicitly to find ^dy⁄_dx given x + sin(y) = 1.

• A) -cos(y)
• B) ¹⁄_cos(y)
• C) -¹⁄_cos(y)
• D) -sin(y)

View Answer & Solution

Correct Answer: C

Solution:
1. Differentiate term by term with respect to x:
1 + cos(y) · ^dy⁄_dx = 0
2. Move 1 over to the right hand side:
cos(y) · ^dy⁄_dx = -1
3. Divide by cos(y) to finish solving:
^dy⁄_dx = -¹⁄_cos(y)

Question 4: Find ^dy⁄_dx if x² + 3xy = 10.

• A) -^{2x + 3y}⁄_3x
• B) -^2x⁄₃
• C) ^{2x + 3y}⁄_3x
• D) -^{2x + 3}⁄_3x

View Answer & Solution

Correct Answer: A

Solution:
1. Differentiate term by term. Use the product rule for the 3xy part:
2x + [3x(^dy⁄_dx) + 3y(1)] = 0
2x + 3x(^dy⁄_dx) + 3y = 0
2. Keep the ^dy⁄_dx term on the left side, and move the rest to the right:
3x(^dy⁄_dx) = -2x - 3y
3. Factor out the negative sign on the right side and divide by 3x:
3x(^dy⁄_dx) = -(2x + 3y)
^dy⁄_dx = -^{2x + 3y}⁄_3x

Question 5: Find the slope of the tangent line ^dy⁄_dx for the curve y² - x = 5.

• A) 2y
• B) ¹⁄_2y
• C) -¹⁄_2y
• D) 1

View Answer & Solution

Correct Answer: B

Solution:
1. Differentiate implicitly with respect to x:
2y(^dy⁄_dx) - 1 = 0
2. Move the constant -1 to the right side:
2y(^dy⁄_dx) = 1
3. Divide by 2y to isolate the derivative expression:
^dy⁄_dx = ¹⁄_2y