Let y=(x2+2)^4
Find the differential dy when x=1 and dx=0.5
Find the differential dy when x=1 and dx=0.01
use the chain rule
dy/dx = 4*(x^2+2)^3*2x
so, dy = 8x(x^2+2)^3 dx
Now plug in your numbers
Well, if y = (x^2 + 2)^4, then we can find the differential dy using the power rule for differentiation. Let's put on our mathematical clown shoes and give it a try!
When x = 1 and dx = 0.5, we need to find dy. Are you ready? Here we go!
First, let's find dy when x = 1 and dx = 0.5:
dy = 4 * (x^2 + 2)^3 * 2x * dx
Substituting x = 1 and dx = 0.5:
dy = 4 * (1^2 + 2)^3 * 2(1) * 0.5
dy = 4 * (3)^3 * 2 * 0.5
dy = 4 * 27 * 2 * 0.5
dy = 4 * 27 * 1
dy = 108
So, when x = 1 and dx = 0.5, the differential dy is 108.
Now, let's find dy when x = 1 and dx = 0.01. Are you ready for another math-tastic adventure?
When x = 1 and dx = 0.01:
dy = 4 * (x^2 + 2)^3 * 2x * dx
Substituting x = 1 and dx = 0.01:
dy = 4 * (1^2 + 2)^3 * 2(1) * 0.01
dy = 4 * (3)^3 * 2 * 0.01
dy = 4 * 27 * 2 * 0.01
dy = 4 * 27 * 0.02
dy = 2.16
So, when x = 1 and dx = 0.01, the differential dy is 2.16.
To find the differential dy, we can use the chain rule.
The chain rule states that if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.
In this case, our function y = (x^2 + 2)^4.
Let's find the differential dy when x = 1 and dx = 0.5:
Step 1: Find dy/du
To find dy/du, we differentiate y with respect to u.
Let u = x^2 + 2
Using the power rule, dy/du = 4(u^3)
Step 2: Find du/dx
To find du/dx, we differentiate u with respect to x.
u = x^2 + 2
Using the power rule, du/dx = 2x
Step 3: Calculate dy/dx
Now, we can use the chain rule formula:
dy/dx = dy/du * du/dx
dy/dx = 4(u^3) * 2x
Substituting u = x^2 + 2 and the given values x = 1 and dx = 0.5:
dy/dx = 4((1^2 + 2)^3) * 2(1)
dy/dx = 4(3^3) * 2
dy/dx = 4(27) * 2
dy/dx = 216
Therefore, the differential dy when x = 1 and dx = 0.5 is 216.
Similarly, let's find the differential dy when x = 1 and dx = 0.01:
Substituting the given values x = 1 and dx = 0.01:
dy/dx = 4((1^2 + 2)^3) * 2(1)
dy/dx = 4(3^3) * 2
dy/dx = 4(27) * 2
dy/dx = 216
Therefore, the differential dy when x = 1 and dx = 0.01 is also 216.
To find the differential dy, we can use the chain rule of differentiation. The chain rule states that if we have a function of the form y = f(g(x)), then the derivative of y with respect to x can be found by multiplying the derivative of f with respect to g by the derivative of g with respect to x.
In this case, we have y = (x^2 + 2)^4. Let's differentiate y with respect to x using the chain rule:
Step 1: Find the derivative of y with respect to g. We have f(g) = g^4, so the derivative of f with respect to g is 4g^3.
Step 2: Find the derivative of g with respect to x. Here, g(x) = x^2 + 2. The derivative of g with respect to x is 2x.
Step 3: Multiply the derivatives obtained in steps 1 and 2.
So, dy/dx = (4(x^2 + 2)^3) * (2x).
Now, let's find the value of dy when x = 1 and dx = 0.5.
Substitute x = 1 into dy/dx = (4(x^2 + 2)^3) * (2x), we get:
dy/dx = (4(1^2 + 2)^3) * (2 * 1)
= (4(1 + 2)^3) * (2)
= (4(3)^3) * (2)
= (4 * 27) * 2
= 108 * 2
= 216
Therefore, when x = 1 and dx = 0.5, the differential dy is equal to 216.
Now, let's find the differential dy when x = 1 and dx = 0.01.
Again, substitute x = 1 into dy/dx = (4(x^2 + 2)^3) * (2x):
dy/dx = (4(1^2 + 2)^3) * (2 * 1)
= (4(1 + 2)^3) * (2)
= (4(3)^3) * (2)
= (4 * 27) * 2
= 108 * 2
= 216
Therefore, when x = 1 and dx = 0.01, the differential dy is also equal to 216.