derivative of loga(u)
To find the derivative of \(\log_a(u)\), you can use the logarithmic differentiation technique
To find the derivative of \(\log_a(u)\), you can use the logarithmic differentiation technique.
Using the change of base formula, we can rewrite \(\log_a(u)\) in terms of natural logarithms as:
\(\log_a(u) = \frac{\ln(u)}{\ln(a)}\)
Now, let’s differentiate both sides of the equation with respect to \(u\):
\(\frac{d}{du}(\log_a(u)) = \frac{d}{du}\left(\frac{\ln(u)}{\ln(a)}\right)\)
Using the quotient rule of differentiation on the right side, we have:
\(\frac{d}{du}(\log_a(u)) = \frac{\frac{d}{du}(\ln(u))\cdot\ln(a) – \ln(u)\cdot\frac{d}{du}(\ln(a))}{(\ln(a))^2}\)
The derivative of \(\ln(u)\) with respect to \(u\) is simply \(\frac{1}{u}\), and since \(\ln(a)\) is a constant, the derivative of \(\ln(a)\) with respect to \(u\) is zero. This simplifies our expression to:
\(\frac{d}{du}(\log_a(u)) = \frac{\frac{1}{u}\cdot\ln(a)}{(\ln(a))^2}\)
Simplifying further, we have:
\(\frac{d}{du}(\log_a(u)) = \frac{1}{u(\ln(a))}\)
Therefore, the derivative of \(\log_a(u)\) with respect to \(u\) is \(\frac{1}{u(\ln(a))}\).
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