What is the derivative of $f (x) = x \cdot {log}_{5} (x)$ $f(x)=x*log_5(x)$ ?

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What is the derivative of $f (x) = x \cdot {log}_{5} (x)$ $f(x)=x*log_5(x)$ ?

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Answer 1 · 2014-08-06T14:51:19

When you're differentiating an exponential with a base other than $e$ $e$ , use the change-of-base rule to convert it to natural logarithms:

$f (x) = x \cdot \frac{ln x}{ln 5}$ $f(x) = x * lnx/ln5$

Now, differentiate, and apply the product rule:

$\frac{d}{d x} f (x) = \frac{d}{d x} [x] \cdot \frac{ln x}{ln 5} + x \cdot \frac{d}{d x} [\frac{ln x}{ln 5}]$ $d/dxf(x) = d/dx[x] * lnx/ln5 + x * d/dx[lnx/ln5]$

We know that the derivative of $ln x$ $ln x$ is $\frac{1}{x}$ $1/x$ . If we treat $\frac{1}{ln 5}$ $1/ln5$ as a constant, then we can reduce the above equation to:

$\frac{d}{d x} f (x) = \frac{ln x}{ln 5} + \frac{x}{x ln 5}$ $d/dxf(x) = lnx/ln5 + x/(xln5)$

Simplifying yields:

$\frac{d}{d x} f (x) = \frac{ln x + 1}{ln 5}$ $d/dxf(x) = (lnx+1)/ln5$

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What is the derivative of $f (x) = x \cdot {log}_{5} (x)$ $f(x)=x*log_5(x)$ ?

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What is the derivative of $f (x) = x \cdot {log}_{5} (x)$ $f(x)=x*log_5(x)$ ?