\begin{align}(a^x)’&=a^x\cdot \ln a\\(e^x)’&=e^x\\(\ln x)’&=\frac{1}{x}\\(\sin x)’&=\cos x\\(\cos x)’&=-\sin x\end{align}
Oppgave 1
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\begin{align*}f(x)&=e^{x}+\ln(x)\\f'(x)&=e^x+\frac{1}{x}\end{align*}
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\begin{align*}f(x)&=3\cdot\ln(x)\\f'(x)&=\frac{3}{x}\end{align*}
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\begin{align*}f(x)&=\frac{-4}{\sqrt{x}}\\&=-4\cdot x^{-1/2}\\f'(x)&=(-4)\cdot(-\frac{1}{2})\cdot x^{-\frac{1}{2}-1}\\&=2x^{-\frac{3}{2}}\\&=\frac{2}{x\sqrt{x}}\end{align*}
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\begin{align*}f(x)&=2\sqrt{x^3}\\&=2x^{\frac{3}{2}}\\f'(x)&=2\cdot \frac{3}{2}\cdot x^{\frac{3}{2}-1}\\&=3x^{\frac{1}{2}}\\ &=3\sqrt{x}\end{align*}
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\begin{align*}f(x)&=x^3-2x+\ln{(2x)}\\f'(x)&=3x^2-2+\frac{2}{2x}\\&=3x^2-2+\frac{1}{x}\end{align*}
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\begin{align*}f(x)&=x^4-4x+17\\f'(x)&=4x^3-4\\&=4(x^3-1)\end{align*}
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\begin{align*}f(x)&=\sqrt{x}-x-x^2\\f'(x)&=\frac{1}{2\sqrt{x}}-1-2x\end{align*}
Oppgave 2
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\begin{align*}f(x)&=x\sqrt{x}+x\\&=x^{\frac{3}{2}}+x\\f'(x)&=\frac{3}{2}x^{\frac{1}{2}}+1\\&=\frac{3\sqrt{x}}{2}+1\end{align*}
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\begin{align*}f(x)&=\ln{x^4}\\&=4\ln{x}\\f'(x)&=4\cdot \frac{1}{x}\\&=\frac{4}{x}\end{align*}
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\begin{align*}f(x)&=\ln{x^3}-1\\&=3\ln x -1\\f'(x)&=\frac{3}{x}\end{align*}
Oppgave 3