Question

Let (y=(x^2+1)^5). Find the differential (dy) when (x=4) and (dx=0.1). Find the differential (dy) when (x=4) and (dx=0.05).

a) (dy = 0.1)
b) (dy =0.1)
c) (dy=0.05)
d) (dy =0.05)

Answer 1

To find the differential (dy) when (x=4) and (dx=0.1) or (dx=0.05), we need to find the derivative of the given function y=(x^2+1)^5 and then multiply it by the given differential dx.

Step-by-step explanation:

To find the differential (dy) when (x=4) and (dx=0.1), we need to find the derivative of the given function y=(x^2+1)^5. Taking the derivative, we have:

dy/dx = 5(x^2+1)^4 * 2x

Substituting x=4 into the derivative expression, we get dy/dx = 5(4^2+1)^4*2(4) = 5(17)^4*8 ≈ 9,685,760

Now, to find the differential dy, we multiply the derivative by the differential dx. Therefore, dy = dy/dx * dx = 9,685,760 * 0.1 = 968,576

For dx=0.05, we repeat the process using the same derivative expression and substituting dx=0.05. We get dy/dx = 9,685,760 * 0.05 = 484,288. Therefore, dy = dy/dx * dx = 484,288 * 0.05 = 24,214.4