Whenever possible, rewrite a function so it looks good *before* taking its derivative.

Any logarithm log_{b }a can be written as

This allows us to find derivatives of logarithms in bases besides *e*.

We also know enough now to find derivatives of things like

*f*(*x*) = *ln* (*x*^{2}).

We can bring down the exponent to find

*f*(*x*) = 2*ln* x,

and we know how to find the derivative of this:

## Practice:

Find the derivative of *f*(*x*) = log_{10}* x*. | |

We can rewrite this logarithm as Since (*ln* 10) is a constant, the function *f* is a constant multiple of *ln* *x*: We know how to find the derivative of this function:
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Find the derivative of *f*(*x*) = log_{4 }*x*^{2}.
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First, simplify the original function. We need to bring the exponent down in front, split up the logarithm, and combine all the constants into one: Or, we could instead split up the logarithm first and then bring down the exponent (the order of these two operations doesn't matter): Now we have a nicely rewritten function: We've done an awful lot of work and haven't even started taking the derivative yet - but now, taking the derivative will be a piece of cake. The function *f* is a constant multiplied by *ln* *x*, therefore We could make this even tidier by recognizing that
*ln* 4 = *ln* (2^{2}) = 2*ln* 2, therefore
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Find the derivative of the function.

Answer

For each of these functions, we need to do two things. First, rewrite the function to look better. Second, take the derivative of the better looking function.

- First, we rewrite the original function:

Next, we take the derivative:

Find the derivative of the function.

Answer

- This problem is exactly the same as the previous one, but with 2 everywhere instead of 9. First, we rewrite the original function:

Next, we take the derivative:

Find the derivative of the function.

Answer

*h*(*x*) = *ln* *x*^{{10}} = 10*ln* *x*

Then we take the derivative:

Find the derivative of the function.

Answer

- This one requires some heavy-duty rewriting.

Now we can take the derivative:

Find the derivative of the function.

Answer

- This one also requires a fair bit of rewriting.

And now we take the derivative: