Let y = ((x^(2))+4)^5.

Find the differential dy when x = 4 and dx = 0.4
Find the differential dy when x = 4 and dx = 0.05

To find the differential dy, we can use the chain rule for differentiation. The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x), where f'(x) represents the derivative of f with respect to x, and g'(x) represents the derivative of g with respect to x.

In this case, let's break down the expression y = ((x^(2))+4)^5 into two functions: f(u) = u^5 and g(x) = x^2 + 4. Notice that f(u) is a power function and g(x) is a polynomial.

First, let's find the derivative of g(x) with respect to x, which is g'(x).

g(x) = x^2 + 4
Using the power rule, the derivative is: g'(x) = 2x.

Next, let's find the derivative of f(u) with respect to u, which is f'(u).

f(u) = u^5
Using the power rule, the derivative is: f'(u) = 5u^4.

Now, we can substitute these derivatives into the chain rule formula to find dy/dx.

dy/dx = f'(g(x)) * g'(x)
= 5(x^2 + 4)^4 * 2x

To find the differential dy, we can substitute the values of x and dx into dy/dx and calculate the result.

When x = 4 and dx = 0.4:
dy = dy/dx * dx
= 5(4^2 + 4)^4 * 2(4) * 0.4
≈ 2.570*10^9

When x = 4 and dx = 0.05:
dy = dy/dx * dx
= 5(4^2 + 4)^4 * 2(4) * 0.05
≈ 321,250