d/dx[uⁿ]
nuⁿ⁻¹u′
To differentiate u^n with respect to x, we need to use the chain rule of differentiation because u is a function of x and n is a constant. Therefore, we have:
d/dx[u^n] = d/du[u^n] * d/dx[u]
The first term on the right-hand side represents the derivative of u^n with respect to u, which is given by:
d/du[u^n] = n u^(n-1)
The second term represents the derivative of u with respect to x, which is given by du/dx. Hence, we can combine these two terms to obtain the final answer:
d/dx[u^n] = d/du[u^n] * du/dx
= n u^(n-1) * du/dx
Therefore, the derivative of u^n with respect to x is n u^(n-1) times the derivative of u with respect to x.
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