Question

Write The Function In The Form Y = f(U) And U = g(X). Then Find dy/dx As A Function Of X. Y=(-2x+8)⁵

Answer 1

The function y=(-2x+8)⁵ can be represented as y=f(u) with f(u)=u⁵ and u=g(x) with g(x)=-2x+8. Using the chain rule, the derivative dy/dx is calculated as -10(-2x+8)⁴.

Step-by-step explanation:

To express the function y=(-2x+8)⁵ in the form of y = f(u) and u = g(x), let's introduce a substitution for the inner function -2x+8. First, set u = -2x + 8. Then, the function can be rewritten as y = u⁵. This gives us the two functions we're looking for: f(u) = u⁵ and g(x) = -2x + 8.

To find dy/dx as a function of x, we use the chain rule which states that dy/dx = (dy/du) × (du/dx). Taking the derivatives of f(u) and g(x) separately, we have dy/du = 5u⁴ and du/dx = -2. Plug these values into the chain rule to get dy/dx = 5u⁴ × (-2) = -10u⁴. Since u = -2x + 8, we can substitute u back into dy/dx to get dy/dx = -10(-2x + 8)⁴.

Answer 2

`\( (dy)/(dx) \) as a function of \( x \) for the given function \( y = (-2x + 8)^5 \)`is:

`\[ (dy)/(dx) = -160(x - 4)^4 \]`

To express the function
`\( y = (-2x + 8)^5 \) in the form \( y = f(u) \) and \( u = g(x) \), and then find \( (dy)/(dx) \) as a function of \( x \),`we follow these steps:

Step 1: Define u as a Function of x (i.e., u = g(x)

Let's define u such that it encapsulates the inner function of y . In this case, we set:

`\[ u = -2x + 8 \]`

`\[ \implies u = g(x) = -2x + 8 \]`

Step 2: Express y as a Function of u (i.e., y = f(u) )

Now express y in terms of u :

`\[ y = u^5 \]`

`\[ \implies y = f(u) = u^5 \]`

Step 3: Find
`\( (du)/(dx) \) and \( (dy)/(du) \)`

Now, differentiate( u = g(x) with respect to x , and y = f(u) with respect to u :

-
`\( (du)/(dx) = (d)/(dx)(-2x + 8) \)`

-
`\( (dy)/(du) = (d)/(du)(u^5) \)`

Step 4: Apply the Chain Rule to Find
`\( (dy)/(dx) \)`

The chain rule states that
`\( (dy)/(dx) = (dy)/(du) * (du)/(dx) \). We'll calculate \( (dy)/(dx) \)` using the derivatives from Step 3.

Let's perform these calculations:

It appears that there was an error in calculating the derivative of y with respect to u . This is because u was not correctly expressed as a symbolic variable in the context of the derivative calculation. Let's correct this and recalculate
`\( (dy)/(dx) \):`

Corrected Calculation

1. Calculate
`\( (du)/(dx) \):`

`\[ (du)/(dx) = (d)/(dx)(-2x + 8) \]`

2. Calculate
`\( (dy)/(du) \)`with u as a symbolic variable:

`\[ (dy)/(du) = (d)/(du)(u^5) \]`

3. Apply the Chain Rule:

`\[ (dy)/(dx) = (dy)/(du) * (du)/(dx) \]`

Let's recalculate with the correct approach.

The derivative
`\( (dy)/(dx) \) as a function of \( x \) for the given function \( y = (-2x + 8)^5 \)`is:

`\[ (dy)/(dx) = -160(x - 4)^4 \]`

This result is obtained by correctly applying the chain rule to the function y expressed in terms of u and u expressed in terms of x