Dezembro 23, 2019

Qual é a derivada de $3^{x}$ $3 ^ x$ ?

Responda:

$3^{x} ln 3$ $3^xln3$

Explicação:

Begin by letting $y = 3^{x}$ $y=3^x$

now take the ln of both sides.

$ln y = {ln 3}^{x} \Rightarrow ln y = x ln 3$ $lny=ln3^xrArrlny=xln3$

differentiate $implicitly with respect to x$ $color(blue)"implicitly with respect to x"$

$\Rightarrow \frac{1}{y} \frac{d y}{d x} = ln 3$ $rArr1/y dy/dx=ln3$

$\Rightarrow \frac{d y}{d x} = y ln 3$ $rArrdy/dx=yln3$

now y = $3^{x} \Rightarrow \frac{d y}{d x} = 3^{x} ln 3$ $3^xrArrdy/dx=3^xln3$

This result can be $generalised$ $color(blue)"generalised"$ as follows.

$color(red)(bar(ul(|color(white)(a/a)color(black)(d/dx(a^x)=a^xlna)color(white)(a/a)|)))$