Dezembro 23, 2019

por Cindelyn

Como você diferencia $y = {log x}^{2}$ $y = log x ^ 2$ ?

Responda:

$\frac{d y}{d x} = \frac{2}{x}$ $dy/dx=2/x$

Explicação:

There are 2 possible approaches.

$Approach 1$ $color(blue)"Approach 1"$

differentiate using the $chain rule$ $color(blue)"chain rule"$

$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ ∣ ∣ ∣ \frac{2}{2} \frac{d}{d x} (log (f (x))) = \frac{1}{f (x)} . f' (x) \frac{2}{2} ∣ ∣ ∣ - ---------------------------- -$ $color(red)(bar(ul(|color(white)(2/2)color(black)(d/dx(log(f(x)))=1/(f(x)).f'(x))color(white)(2/2)|)))$

$y = log (x^{2})$ $y=log(x^2)$

$\Rightarrow \frac{d y}{d x} = \frac{1}{x^{2}} . \frac{d}{d x} (x^{2}) = \frac{1}{x^{2}} \times 2 x = \frac{2}{x}$ $rArrdy/dx=1/x^2.d/dx(x^2)=1/x^2 xx2x=2/x$

$Approach 2$ $color(blue)"Approach 2"$

Using the $law of logs$ $color(blue)"law of logs"$ then differentiate.

$Reminder ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ ∣ ∣ ∣ \frac{2}{2} {log x}^{n} = n log x \frac{2}{2} ∣ ∣ ∣ - ---------------- -$ $color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(logx^n=nlogx)color(white)(2/2)|)))$

$y = {log x}^{2} = 2 log x$ $y=logx^2=2logx$

$\Rightarrow \frac{d y}{d x} = 2 \times \frac{1}{x} = \frac{2}{x}$ $rArrdy/dx=2xx 1/x=2/x$