Math Problem Statement

Find dy/dx

Find dy/dx for the equation ln(x^2 + √y) = sin(xy^2)

Solution

import symbols, Function, diff, log, sin, sqrt

Define the variables and the function

x, y = symbols('x y') f = log(x2 + sqrt(y)) - sin(x * y2)

Compute the derivative of y with respect to x (dy/dx) implicitly

dy_dx = diffThe derivative dydx\frac{dy}{dx} of the given equation ln(x2+y)=sin(xy2)\ln(x^2 + \sqrt{y}) = \sin(xy^2) is:

dydx=2xy2(x2+y)cos(xy2)x2+y\frac{dy}{dx} = \frac{2x - y^2(x^2 + \sqrt{y})\cos(xy^2)}{x^2 + \sqrt{y}}

Would you like a detailed step-by-step explanation or have any specific questions?

Here are 5 related questions you might explore:

  1. How does implicit differentiation work in solving such problems?
  2. What role does the chain rule play in implicit differentiation?
  3. How would this derivative behave for specific values of xx and yy?
  4. How can you verify this derivative through numerical computation?
  5. How does the presence of the square root and trigonometric terms influence the complexity?

Tip: Always isolate yy' (or dydx\frac{dy}{dx}) clearly when solving

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Math Problem Analysis

Mathematical Concepts

Implicit Differentiation
Logarithmic Functions
Trigonometric Functions
Chain Rule

Formulas

d/dx[ln(u)] = 1/u * du/dx
d/dx[sin(u)] = cos(u) * du/dx
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)

Theorems

Chain Rule
Properties of Logarithms

Suitable Grade Level

Grades 11-12 or Early College