Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

ddx(yx+xy=−2)\frac{d}{dx}\left(\frac{y}{x}+xy=-2\right)

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Multiplication with rational exponentsright chevron icon

Full solution

Q. ddx(yx+xy=−2)\frac{d}{dx}\left(\frac{y}{x}+xy=-2\right)
  1. Apply Quotient Rule: We are given the equation (yx)+xy=−2(\frac{y}{x}) + xy = -2 and we need to find its derivative with respect to xx. To do this, we will apply the rules of differentiation to each term separately.
  2. Apply Product Rule: First, let's differentiate the term (y/x)(y/x). Since yy is a function of xx, we need to apply the quotient rule which is (v(u′−u(v′))/v2(v(u' - u(v')) / v^2, where u=yu = y and v=xv = x. The derivative of yy with respect to xx is dy/dxdy/dx, and the derivative of xx with respect to xx is yy11. So, the derivative of (y/x)(y/x) is yy33.
  3. Differentiate Constant Term: Next, we differentiate the term xyxy. This is a product of two functions, so we use the product rule which is u′v+uv′u'v + uv', where u=xu = x and v=yv = y. The derivative of xx with respect to xx is 11, and the derivative of yy with respect to xx is dydx\frac{dy}{dx}. So, the derivative of xyxy is u′v+uv′u'v + uv'11 which simplifies to u′v+uv′u'v + uv'22.
  4. Combine Derivatives: Now, we differentiate the constant term −2-2. The derivative of any constant with respect to xx is 00.
  5. Simplify Differential Equation: Combining the derivatives from the previous steps, we get the derivative of the entire equation:\newlineddx[yx+xy]=xdydx−yx2+y+xdydx=0\frac{d}{dx}\left[\frac{y}{x} + xy\right] = \frac{x\frac{dy}{dx} - y}{x^2} + y + x\frac{dy}{dx} = 0.
  6. Clear Fraction: We now have a differential equation that we can simplify: \newline(xdydx−y)/x2+y+xdydx=0(x\frac{dy}{dx} - y) / x^2 + y + x\frac{dy}{dx} = 0.\newlineMultiplying through by x2x^2 to clear the fraction, we get:\newlinexdydx−y+x2y+x3dydx=0x\frac{dy}{dx} - y + x^2y + x^3\frac{dy}{dx} = 0.
  7. Combine Like Terms: Combine like terms to simplify the equation further:\newlinexdydx+x3dydx=y−x2yx\frac{dy}{dx} + x^3\frac{dy}{dx} = y - x^2y.\newlineFactor out dydx\frac{dy}{dx} on the left side:\newlinedydx(x+x3)=y(1−x2)\frac{dy}{dx} (x + x^3) = y(1 - x^2).
  8. Factor Out dy/dx: Now, we can solve for dydx\frac{dy}{dx} by dividing both sides by (x+x3)(x + x^3):dydx=y(1−x2)(x+x3).\frac{dy}{dx} = \frac{y(1 - x^2)}{(x + x^3)}.
  9. Solve for dydx\frac{dy}{dx}: We have found the derivative of the given equation with respect to xx, which is dydx=y(1−x2)(x+x3)\frac{dy}{dx} = \frac{y(1 - x^2)}{(x + x^3)}.

More problems from Multiplication with rational exponents

Question
Simplify.\newline811281^{\frac{1}{2}}\newline_____________
Get tutor helpright-arrow

Posted 9 months ago

Question
Simplify.\newline(−32)1/5(-32)^{1/5}\newline______
Get tutor helpright-arrow

Posted 9 months ago

Question
Simplify. Assume all variables are positive.\newlinew43w73\frac{w^{\frac{4}{3}}}{w^{\frac{7}{3}}}\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______
Get tutor helpright-arrow

Posted 5 months ago

Question
Simplify. Assume all variables are positive.\newline(2r13)8(2r^{\frac{1}{3}})^8\newline\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______
Get tutor helpright-arrow

Posted 5 months ago

Question
Simplify. Assume all variables are positive.\newlinev74⋅v94v^{\frac{7}{4}} \cdot v^{\frac{9}{4}}\newline\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______\newline
Get tutor helpright-arrow

Posted 5 months ago

Question
Simplify.\newline114⋅1141^{\frac{1}{4}} \cdot 1^{\frac{1}{4}}\newline______
Get tutor helpright-arrow

Posted 5 months ago

Question
Simplify. \newline32(−1/5)32^{(-1/5)}
Get tutor helpright-arrow

Posted 7 months ago

Question
Simplify. Assume all variables are positive.\newlinew32w52\frac{w^{\frac{3}{2}}}{w^{\frac{5}{2}}}\newline\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______\newline
Get tutor helpright-arrow

Posted 9 months ago

Question
Simplify. Assume all variables are positive.\newline(27x)23(27x)^{\frac{2}{3}}\newline\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______
Get tutor helpright-arrow

Posted 8 months ago

Question
Simplify. Assume all variables are positive.\newlinex14x114\frac{x^{\frac{1}{4}}}{x^{\frac{11}{4}}}\newline\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______\newline
Get tutor helpright-arrow

Posted 7 months ago

View all (744)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant