Question

Найти производную! Помогите

Answer 1

$1) \; y=3^{2x^2}\\\\y'=\frac{d}{dx} (3^{2x^2})\\\\y' =\frac{d}{dg}(9^g) \cdot \frac{d}{dx}(x^2) \; | \; g=x^2\\\\y'=In(9) \cdot 9^{x^2} \cdot 2x\\\\y'=4In(3) \cdot 9^{x^2} \cdot x$

$2) \; y =e^{-2x}\\\\y' = \frac{d}{dx}(e^{-2x})\\\\y'= \frac{d}{dg}(e^g) \cdot \frac{d}{dx}(-2x) \; | \; g=-2x\\\\y'= e^{-2x} \cdot (-2)\\\\y'= -\frac{2}{e^{2x}}$

$3) \; y= 2^{\sqrt{x}}\\\\y'=\frac{d}{dx}(2^{\sqrt{x}})\\\\y' = \frac{d}{dg}(2^g) \cdot \frac{d}{dx}(\sqrt{x}) \; | \; g=\sqrt{x}\\\\y' = In(2) \cdot 2^{\sqrt{x}} \cdot \frac{1}{2 \sqrt{x}} \\\\y'= \frac{In(2) \cdot 2^{\sqrt{x}-1}}{\sqrt{x}}$

$4) \; y=3^{In(x)}\\\\y' =\frac{d}{dx}(3^{In(x)})\\\\y'=\frac{d}{dg}(3^g) \cdot \frac{d}{dx}(In(x)) \; | \; g=In(x)\\\\y'=In(3) \cdot 3^{In(x)} \cdot \frac{1}{x}\\\\y'= \frac{In(3) \cdot 3^{In(x)}}{x}$

$5) \; y=e^x+1\\\\y'=\frac{d}{dx}(e^x+1)\\\\y' = \frac{d}{dx}(e^x)+\frac{d}{dx}(1)\\\\y'=e^x+0\\\\y'=e^x$