41 lines
837 B
TeX
41 lines
837 B
TeX
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\[f(t) = \frac{t^2 -1}{t+1} + 6t^{1/3} + \sqrt{\sin t} + 4^t\]
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Jeg deriverer funksjonen ledd for ledd
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Ledd 1:
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\begin{align*}
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\frac{d}{dt} \frac{t^2-1}{t+1} &= \frac{d}{dt}\frac{(t+1)(t-1)}{t+1} \\
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&= \frac{d}{dt} t-1 \\
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&= 1
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\end{align*}
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Ledd 2:
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\begin{align*}
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\frac{d}{dt} 6t^{1/3} &= \frac{1}{3} \cdot 6t^{(1/3 - 1)} \\
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&= 2t^{-2/3}\\
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&= \frac{2}{\sqrt[3]{t^2}}
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\end{align*}
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Ledd 3:
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\[ \frac{d}{dt} \sqrt{\sin t} \]
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$u = \sin t$
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\begin{align*}
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\frac{dy}{dt} &= \frac{dy}{du} \cdot \frac{du}{dt} \\
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&= \frac{d}{du} \sqrt{u} \cdot \frac{d}{dt} \sin t \\
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&= \frac{1}{2\sqrt{u}} \cdot \cos t \\
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&= \frac{\cos t}{2\sqrt{\sin t}}
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\end{align*}
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Ledd 4:
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\begin{align*}
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\frac{d}{dt} 4^t = 4^t ln(t) \\
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\end{align*}
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\[\frac{df}{dx} = 1 + \frac{2}{\sqrt[3]{t^2}} + \frac{\cos t}{2\sqrt{\sin t}} + 4^t ln(t)\]
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