d x a ? dx We now want to learn to diﬀerentiate any exponential ax . There are two roughly equivalent methods we can use: Method 1: Convert ax to something with base e and use the chain rule. Because ln x is the inverse function to ex we can rewrite a as eln(a) . Thus: � �x ax = eln(a) = ex ln(a) That looks like it might be tricky to diﬀerentiate. Let’s work up to it: d x e dx d 3x e dx = ex = 3e3x (by the chain rule) Remember, ln(a) is just a constant like 3, not a variable. Therefore: d (ln a)x e dx = (ln a)e(ln a)x or d x a = (ln a)ax dx This is a common type of calculation; you should practice it until you are d kx d x e and dx a or comfortable with it. You may either memorize formulas for dx re-derive them every time you need them. d x Recall that a = M (a) · ax . So ﬁnally we know the value of M (a): dx M (a) = ln(a)

d/dx a^x?

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock