Calculus- Early Transcendentals, 2021a

4.6. Derivatives of Exponential & Logarithmic Functions 145
This is a perfectly good answer, but we can improve it slightly. Since
a = e lna
log a (a) = log a (e lna )=lnalog a e
1 = lnalog a e
1
lna = log a e,
we can replace log a e to get
d
dx log a x = 1
xlna .
You may if you wish memorize the formulas.
Derivative Formulas for a x and log a x
d
dx ax =(lna)a x
and
d
dx log a x = 1
xlna .
Because the “trick” a = e lna is often useful, and sometimes essential, it may be better to remember the
trick, not the formula.
Example 4.35: Derivative of Exponential Function
Compute the derivative of f (x)=2 x .
Solution.
d
dx 2x = d dx (eln2 ) x
= d dx exln2
=
( d
dx xln2 )
e xln2
= (ln2)e xln2 = 2 x ln2
♣
Example 4.36: Derivative of Exponential Function
Compute the derivative of f (x)=2 x2 = 2 (x2) .
Solution.
d
dx 2x2 = d dx ex2 ln2