Math Problem Statement
Given x^5+5xy-y^2=y+7, use implicit differentiation to find dy/dx. dy/dx=?
Solution
We are given the equation:
To find , we'll use implicit differentiation, treating as a function of (i.e., ).
Step 1: Differentiate both sides with respect to
Left-hand side:
- The derivative of with respect to is .
- The derivative of involves the product rule:
- The derivative of using the chain rule:
So, differentiating the left-hand side, we get:
Right-hand side:
- The derivative of is .
- The derivative of the constant is .
So, differentiating the right-hand side, we get:
Step 2: Set up the equation
Now, equate the derivatives from both sides:
Step 3: Solve for
First, group all the terms with on one side:
Factor out on the right-hand side:
Finally, solve for :
Thus, the derivative is:
Would you like further details or have any questions?
Here are 5 related questions for you to explore:
- How does implicit differentiation differ from explicit differentiation?
- Can you find for the same equation?
- What would change if the equation involved a trigonometric function like ?
- How does the chain rule help in differentiating composite functions?
- What is the significance of the product rule in calculus?
Tip: When differentiating implicitly, always treat as a function of , and apply the chain rule when differentiating terms involving .
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Chain Rule
Product Rule
Algebra
Formulas
Implicit differentiation of x^n
Product rule: d(uv)/dx = u'v + uv'
Chain rule: d(f(y))/dx = f'(y) * dy/dx
Theorems
Chain Rule
Product Rule
Suitable Grade Level
Grades 11-12, Calculus I