Let y = sqrt(7-x)
Find the differential dy when x= 1 and dx = 0.4
Find the differential dy when x= 1 and dx = 0.02
well,
dy = -1 / 2√(7-x) dx
So, just plug in your numbers
To find the differential dy, we need to use the derivative of the function y = √(7-x).
Step 1: Find the derivative of y with respect to x.
The derivative of y with respect to x can be found using the chain rule. The chain rule states that for a function f(g(x)), the derivative is equal to f'(g(x)) * g'(x).
In this case, the function f(u) = √u, and g(x) = 7-x.
First, find the derivative of f(u) with respect to u:
f'(u) = 1 / (2√u)
Next, find the derivative of g(x) with respect to x:
g'(x) = -1
Using the chain rule, the derivative of y with respect to x is:
dy/dx = f'(g(x)) * g'(x)
= (1 / (2√(7-x))) * (-1)
= -1 / (2√(7-x))
Step 2: Substitute the given values of x and dx into the derivative equation.
For x = 1 and dx = 0.4:
dy/dx = -1 / (2√(7-1))
= -1 / (2√6)
≈ -0.204
For x = 1 and dx = 0.02:
dy/dx = -1 / (2√(7-1))
= -1 / (2√6)
≈ -0.204
Therefore, for both cases, when x=1 and dx=0.4 or dx=0.02, the differential dy is approximately -0.204.
To find the differential dy, we can use the derivative of y with respect to x. In this case, since y is defined as y = sqrt(7-x), we need to find dy/dx.
Let's find the derivative dy/dx of y = sqrt(7-x):
Step 1: Rewrite the expression using the exponent form.
y = (7-x)^(1/2)
Step 2: Apply the power rule of differentiation.
dy/dx = (1/2) * (7-x)^(-1/2) * (-1)
Step 3: Simplify the derivative expression.
dy/dx = -1/2(7-x)^(-1/2)
Now that we have the derivative dy/dx, we can find the differential dy using the formula:
dy = dy/dx * dx
Let's calculate the differential dy for the given values of x and dx:
When x = 1 and dx = 0.4:
dy = (-1/2)(7-1)^(-1/2) * 0.4
dy = (-1/2)(6)^(-1/2) * 0.4
dy ≈ -0.600
When x = 1 and dx = 0.02:
dy = (-1/2)(7-1)^(-1/2) * 0.02
dy = (-1/2)(6)^(-1/2) * 0.02
dy ≈ -0.030
Therefore, when x = 1 and dx = 0.4, the differential dy is approximately -0.600. When x = 1 and dx = 0.02, the differential dy is approximately -0.030.