The derivative of y with respect to x
Consider the following for the two (02) items that follow : Let (x+y)^{p+q}=x^{p}y^{q}, where p, q are positive integers.
- A. depends on p only
- B. depends on q only
- C. depends on both p and q
- D. is independent of both p and q ✓
Correct Answer: D. is independent of both p and q
Explanation
Taking natural log on both sides: (p+q)\ln(x+y) = p\ln x + q\ln y. Differentiating wrt x: \frac{p+q}{x+y}(1 + \frac{dy}{dx}) = \frac{p}{x} + \frac{q}{y}\frac{dy}{dx}. Rearranging gives \frac{dy}{dx}(\frac{p+q}{x+y} - \frac{q}{y}) = \frac{p}{x} - \frac{p+q}{x+y}. Simplifying both sides yields \frac{dy}{dx}(\frac{py-qx}{y(x+y)}) = \frac{py-qx}{x(x+y)} \implies \frac{dy}{dx} = \frac{y}{x}. Thus, the derivative depends on neither p nor q.
Related questions on Calculus
- Let z=[y] and y=[x]-x, where [.] is the greatest integer function. If x is <strong>NOT</strong> an integer but positive, then what i...
- If f(x)=4x+1 and g(x)=kx+2 such that fog(x)=gof(x), then what is the value of k?
- What is the <strong>MINIMUM</strong> value of the function f(x)=\log_{10}(x^{2}+2x+11)?
- What is \int(x^{x})^{2}(1+\ln x)\,dx equal to ?
- What is \int e^{x}\{1+\ln x+x\ln x\}\,dx equal to?