The Chain Rule | Finding the Derivative of u^n with Respect to x

d/dx (u^n)

To find the derivative of u raised to the power of n, where u is a function of x, we can use the chain rule

To find the derivative of u raised to the power of n, where u is a function of x, we can use the chain rule. The chain rule states that for a composition of functions, the derivative can be found by multiplying the derivative of the outer function with the derivative of the inner function.

In this case, let’s assume that u is a differentiable function of x. We want to find the derivative of (u^n) with respect to x, denoted as d/dx (u^n).

Using the chain rule, we consider u^n as the outer function and u as the inner function. We can express (u^n) as (u^(n-1)) * u.

Now, let’s differentiate both sides with respect to x:

d/dx (u^n) = d/dx ((u^(n-1)) * u)

Applying the product rule on the right hand side, we have:

d/dx (u^n) = d/dx (u^(n-1)) * u + (u^(n-1)) * d/dx(u)

The first term on the right-hand side involves differentiating (u^(n-1)) with respect to x while treating u^(n-1) as a constant. This can be obtained by multiplying the derivative of (u^(n-1)) with respect to u (which is (n-1)*u^(n-2)) by du/dx.

Similarly, the second term on the right-hand side involves differentiating u with respect to x while treating (u^(n-1)) as a constant, which is simply du/dx.

Combining these terms, we get the derivative of (u^n) as:

d/dx (u^n) = (n-1) * u^(n-2) * du/dx * u + u^(n-1) * du/dx

Simplifying, we can write this as:

d/dx (u^n) = (n * u^(n-1)) * du/dx

So, the derivative of u raised to the power of n, with respect to x, is (n * u^(n-1)) times the derivative of u with respect to x.

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