Math Problem Statement

teach me implicit diffrenciation and give me question

Solution

Implicit Differentiation Explanation

Implicit differentiation is used when a function is not explicitly solved for one variable in terms of another. This typically happens when yy is given as a function of xx, but not in the standard form y=f(x)y = f(x). In such cases, both xx and yy are treated as variables, and we need to apply the chain rule during differentiation.


How it Works:

  1. Differentiate both sides of the equation with respect to xx.

    • Treat yy as a function of xx, meaning every time you differentiate yy, you must multiply by dydx\frac{dy}{dx} (due to the chain rule).
  2. Apply the product rule and chain rule where necessary.

  3. Solve for dydx\frac{dy}{dx}, which represents the derivative of yy with respect to xx.


Example:

Differentiate the following equation implicitly:
x2+y2=25x^2 + y^2 = 25

Solution:

  1. Differentiate both sides with respect to xx: ddx(x2)+ddx(y2)=ddx(25)\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)

  2. Use the power rule on x2x^2 and the chain rule on y2y^2: 2x+2ydydx=02x + 2y \frac{dy}{dx} = 0

  3. Rearrange the equation to solve for dydx\frac{dy}{dx}: 2ydydx=2x2y \frac{dy}{dx} = -2x

  4. Divide by 2y2y: dydx=xy\frac{dy}{dx} = -\frac{x}{y}


Practice Problem:

Use implicit differentiation to find dydx\frac{dy}{dx} for the equation:

x3+y3=6xyx^3 + y^3 = 6xy


Let me know if you want the solution to this problem or have any questions about implicit differentiation!


5 Related Questions:

  1. How does implicit differentiation differ from explicit differentiation?
  2. What is the chain rule, and why is it used in implicit differentiation?
  3. How can implicit differentiation be applied to find the slope of a curve at a point?
  4. Can we use implicit differentiation with trigonometric functions?
  5. What happens if a function involves higher derivatives?

Tip: When dealing with implicit equations, always isolate dydx\frac{dy}{dx} at the end for clarity.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Implicit Differentiation
Differentiation Rules

Formulas

Chain Rule
Product Rule

Theorems

Chain Rule in Differentiation
Product Rule in Differentiation

Suitable Grade Level

Grades 11-12