Derivative of 2^x: How to Calculate

[l33t_V]

Hello all. I would like to know what is the derivative of 2^x and how it is done. Thank you

 

[uart]

Here's a hint, start by expressing it as :

$$2^x = e^{x \ln(2)}$$

 

[l33t_V]

I'll give it a try. But why can't it be solved as we do x^k ? Like k*x'*x^(k-1)

 

[Tac-Tics]

I'll give it a try. But why can't it be solved as we do x^k ? Like k*x'*x^(k-1)

Because that's against the rules.

The rule is

$$\frac{d}{dx}(x^n) = n x^{n-1}$$

The base has to be the variable being differentiated against. The exponent has to be constant with respect to the base.

Luckily, all is not lost. We have lots of rules in calculus. Instead, we can use this one.

$$\frac{d}{dx}(e^{f(x)}) = f'(x) e^{f(x)}$$

This rule can only be used when the base is the Euler number e and when the exponent is a function of x. If we want to use this rule for your problem, we can do a little advanced algebra to change the base (which is what uart suggested).

Some problems in calculus can't be solved exactly, even if they look super simple. For example, if you have the function $$f(x) = x^x$$, you can use NEITHER rule.

 

[lurflurf]

Use the power rule.

$$\frac{d}{dx}u^v=v u^{v-1} \frac{du}{dx}+\log(u) u^v \frac{dv}{dx}$$

Another thread about it.