d/dx(a^u)
a^u(lna)(u’)
To differentiate a function of the form a^u with respect to x, we use the chain rule of differentiation.
Let y = a^u, where a is a constant and u is a function of x.
Then, taking the natural logarithm (ln) of both sides gives:
ln(y) = ln(a^u)
ln(y) = u * ln(a)
Differentiating both sides with respect to x yields:
1/y * dy/dx = d/dx(u * ln(a))
1/y * dy/dx = u’ * ln(a) + u * d/dx(ln(a))
Note that d/dx(ln(a)) = 0 since ln(a) is a constant.
Substituting y = a^u gives:
dy/dx = y * u’ * ln(a)
dy/dx = a^u * u’ * ln(a)
Therefore, the derivative of a^u with respect to x is a^u * u’ * ln(a).
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