Question

What is the differenciation of $2^x$ wrt $x$ ?

Answer 1

We need to find the differentiation of $2^x$.

Apply exponent rule $a^b=e^{bln\left(a\right)}$

$\Rightarrow 2^x=e^{xln\left(2\right)}...\left(1\right)$

Thus we need to find $\frac{d}{dx}\left(e^{x\left(ln\right(2\left)\right)}\right)$

Let $f=e^u.u=xln\left(2\right)$

We know that $\frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}$

Thus $\frac{d}{dx}\left(e^{x\left(ln\right(2\left)\right)}\right)=\frac{d}{du}\left(e^u\right)\cdot \frac{d}{dx}\left(xln\right(2\left)\right)$

$=e^u\cdot ln\left(2\right)$

Substitute back $u=xln\left(2\right)$ we get

$=e^{xln\left(2\right)}\cdot ln\left(2\right)$

$=2^x\cdot ln\left(2\right)by\left(1\right)$

Thus $\frac{d}{dx}\left(2^x\right)=2^x\cdot ln\left(2\right)$