Differentiating exponentials and logarithmic functions
In a nutshell
Being able to differentiate exponentials (for example akx) and logarithms (for example log(kx)) is a useful skill to have, since these functions (especially those involving e and ln) appear in real-life scenarios frequently.
"Natural" functions
Given the following functions for a constant k:
yy=ekx=ln(x)
their respective derivatives are given by:
y=ekx→dxdy=kekx y=ln(x)→dxdy=x1
Consider now the function:
y=ln(kx)
Before taking the derivative, use logarithm rules to re-express this:
yy=ln(kx)=ln(k)+ln(x)
Since ln(k) is just a constant, it follows that the derivative of y=ln(kx) is also:
y=ln(kx)→dxdy=x1
Example 1
Differentiate the following function with respect to x:
f(x)=4e6x−2ln(5x)
Differentiating each term individually gives:
f′(x)f′(x)=6×4e6x−x2=24e6x−x2
General exponential
Consider now the function:
y=ax
where a is a constant. Differentiating this won't be as straightforward as differentiating ex since e is the unique number such that differentiating ex gives itself.
To differentiate, first, using logarithm rules, rewrite this function:
yy=eln(ax)=exln(a)
Since lna is just a constant, it follows from the rule at the top that:
dxdydxdy=ln(a)exln(a)=ln(a)eln(ax)
Thus:
y=ax→dxdy=ln(a)ax
In a very similar way, you can show that akx differentiates to kln(a)akx.
General logarithm
Finally consider the general logarithm:
y=loga(x)
where a is a constant. Again, since this is not the natural logarithm, it will not differentiate as easily as ln(x) does. Instead, it should first be rewritten:
yayln(ay)yln(a)y=loga(x)=x=ln(x)=ln(x)=ln(a)ln(x)
Since ln(a) is just a constant, it follows from the rule at the top that:
y=loga(x)→dxdy=xln(a)1
Note: If a=e, then you have the same result as before, since ln(e)=1.
In a very similar way, you can show that loga(kx) differentiates to xln(a)1.
Example 2
Differentiate the following function with respect to x:
g(x)=32x+log10(4x)
Differentiating each term individually gives:
g′(x)g′(x)g′(x)=2×ln(3)×32x+xln(10)1=2(32x)ln(3)+xln(10)1=9xln(9)+xln(10)1
