Math Problem Statement

dxdy​=2x−1 derivative

Solution

To find the derivative of the equation dxdy=2x1\frac{dx}{dy} = 2x - 1, we can follow these steps:

  1. Interpret the equation: We have dxdy\frac{dx}{dy}, which means we are taking the derivative of xx with respect to yy. The right-hand side is 2x12x - 1.

  2. Differentiate implicitly: Since xx is a function of yy, we need to differentiate both sides with respect to yy, applying the chain rule to xx.

    Differentiating xx with respect to yy will give dxdy\frac{dx}{dy}, and differentiating the right side gives:

    ddy(2x1)=2dxdy\frac{d}{dy}(2x - 1) = 2 \cdot \frac{dx}{dy}

    Therefore, the equation becomes:

    dxdy=2dxdy1\frac{dx}{dy} = 2 \cdot \frac{dx}{dy} - 1

  3. Solve for dxdy\frac{dx}{dy}:

    dxdy2dxdy=1\frac{dx}{dy} - 2 \cdot \frac{dx}{dy} = -1

    Simplifying:

    dxdy=1-\frac{dx}{dy} = -1

    Multiply both sides by 1-1:

    dxdy=1\frac{dx}{dy} = 1

Thus, the derivative of xx with respect to yy is:

dxdy=1\frac{dx}{dy} = 1

Do you want a more detailed explanation or have any further questions?


Here are 5 related questions to explore further:

  1. How would you solve for xx from dxdy=1\frac{dx}{dy} = 1?
  2. What does the equation dxdy=2x1\frac{dx}{dy} = 2x - 1 represent in terms of a differential equation?
  3. How do you apply implicit differentiation to other equations?
  4. What is the role of the chain rule in this differentiation process?
  5. How can this result be interpreted graphically?

Tip: When performing implicit differentiation, always remember to treat any variables (like xx in this case) that are functions of the other variable (here yy) using the chain rule.

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Math Problem Analysis

Mathematical Concepts

Implicit Differentiation
Derivatives
Chain Rule

Formulas

d/dy(2x - 1) = 2 * dx/dy

Theorems

Implicit Differentiation
Chain Rule

Suitable Grade Level

Grades 11-12