Question

QUIZZZ !!!

[tex]\frac{d}{dx} \: \: 4 {x}^{2} - {2x}^{ {x}^{2} } +{( cos \: {45}^{o} ) }^{x} [/tex]
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Answer 1

TUrunan

y = 4x² → y' = 8x

y = (cos 45°)^x

y' = (cos 45°)^x . 1 . ln (cos 45°)

y' = ((1/2 √2)^x) ln (1/√2)

y = x^(x²)

ln y = x² ln x

y' . 1/y = 2x ln x + x² . 1/x

y' . 1/y = 2x ln x + x

y' = 2xy lx x + xy

y' = y(x + 2x ln x)

y' = (x^(x²))(x + 2x ln x)

d(4x² - 2x^(x²) + (cos 45°)^x) / dx

= 8x - 2 (x^(x²))(x + 2x ln x) + ((1/2 √2)^x) ln (1/√2)

= 8x - x[x^(x²)][2 + 4 ln x] - 1/2 ((1/2 √2)^x) ln 2

Answer 2

Verified answer

Penjelasan dengan langkah-langkah:

Cos 45° = [tex]\tt \frac{1}{2}\sqrt{2}[/tex]

[tex]\tt \frac{d}{dx}(4x^2-2x^{x^2}+cos(45^o)^x)\\\\= \frac{d}{dx}(4x^2-2x^{x^2}+(\frac{1}{2}\sqrt{2})^x)\\\\=\frac{d}{dx}(4x^2-2x^{x^2}+\frac{(2^\frac{1}{2})^x}{2^x} )\\ \\ =\frac{d}{dx}(4x^2-2x^{x^2}+2^{\frac{1}{2}x-x}) \to gunakan~teorema~turunan:\frac{d}{dx}(a\times f)=a\times \frac{d}{dx}(f) \\\\=2(4x)- 2(\frac{d}{dx}(x^{x^2}))+\frac{d}{dx}(2^{-\frac{1}{2}x})\to gunakan~logaritma~natural:e^{ln~x}=x[/tex]

[tex]\tt = 8x-2(\frac{d}{dx}((e^{ln(x)})^{(x^2)}))+ln(2)(2^{-\frac{1}{2}x})(-\frac{1}{2})\\\\= 8x-2(\frac{d}{dx}(e^{ln(x)(x^2)}))+ln(2)(2^{-\frac{1}{2}x})(-\frac{1}{2})[/tex]

[tex]\tt =8x-2e^{ln(x)(x^2)}(\frac{1}{x}x^2+ln(x)(2x))+ln(2)(2^{-\frac{1}{2}x})(-\frac{1}{2})\\\\=8x-2e^{ln(x^{(x^2)})}(x+2x(ln(x)))-ln(2)(2^{-\frac{1}{2}x-1})[/tex]

[tex]\tt =8x-2x^{x^2+1}-4x^{x^2+1}ln(x)-ln(2)(2^{-\frac{1}{2}x-1})[/tex]