Question

Find the derivative of w=(t 4 +10) ^92

Answer 1

The derivative of w with respect to t, where w =
`(t^4 + 10)^92, is 368(t^4 + 10)^91 * 4t^3.`

Step-by-step explanation:

The given function w =
`(t^4 + 10)^92` can be expressed as the composition of two functions: u =
`t^4 + 10 and v = u^92`. Applying the chain rule, the derivative of w with respect to t is the product of the derivative of v with respect to u and the derivative of u with respect to t.

Firstly, find the derivative of v with respect to u:

`\[ (dv)/(du) = 92u^(91) \]`

Next, find the derivative of u with respect to t:

`\[ (du)/(dt) = 4t^3 \]`

Now, applying the chain rule:

`\[ (dw)/(dt) = (dv)/(du) * (du)/(dt) \]`

Substitute the derivatives we found earlier:

`\[ (dw)/(dt) = 92(t^4 + 10)^(91) * 4t^3 \]`

Simplify the expression:

`\[ (dw)/(dt) = 368t^3(t^4 + 10)^(91) \]`

So, the final answer is the derivative of w with respect to t is
`368t^3(t^4 + 10)^(91)`. This result represents the rate at which w changes concerning t, considering the power and chain rule applied to the given function.