Question

Find d^2y/dx^2 if y/x^2 = 10

A. 0
B. (2y)/x
C. 20
D. y/x
E. (x-2y)/x

Answer 1

By applying the quotient rule of differentiation to y/x^2 = 10 and setting the result equal to 0, we find that the second derivative dy/dx of y with respect to x is (2y)/x.

Step-by-step explanation:

To find the second derivative, d2y/dx2, of the function given by y/x2 = 10, we first differentiate both sides with respect to x. Assuming y is a function of x (y(y)), we get (d/dx)(y/x2) = (d/dx)(10). The left-hand side requires the use of the quotient rule for differentiation: d/dx(u/v) = (v(du/dx) - u(dv/dx))/v2.

Applying the quotient rule, we have:

• u = y, so du/dx = dy/dx.
• v = x2, so dv/dx = 2x.
• Therefore, d/dx(y/x2) = (x2(dy/dx) - y(2x))/(x2)2 = (x2 dy/dx - 2xy) / x4.

Since the right hand side is simply 0 (the derivative of a constant is 0), we are left with:

x2 dy/dx - 2xy = 0

Now, to find the second derivative, we differentiate both sides once more with respect to x. Again using the quotient rule and collecting terms, we end up with:

d2y/dx2 = (2y)/x

Answer 2

C. 20

Step-by-step explanation:

`\frac{y}{ {x}^(2) } = 10 \\ \\ y = 10 {x}^(2) \\ \\ differentiating \: w.r.t. \: x \: on \: both \: sides \\ \\ (dy)/(dx) = 10 (d)/(dx) {x}^(2) \\ \\ (dy)/(dx) = 10 * 2{x} \\ \\ (dy)/(dx) = 20{x} \\ \\ differentiating \: again\: w.r.t. \: x \: on \: both \: sides \\ \\ (d)/(dx) \bigg((dy)/(dx) \bigg)= 20(d)/(dx) {x} \\ \\ \frac{ {d}^(2) y}{d {x}^(2) } = 20 * 1 \\ \\ \huge \purple {\frac{ {d}^(2) y}{d {x}^(2) } } \red{= } \orange{20}`