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salmanhiro

UTS 2 Matematika IIA ITB 2017 (Bagian A nomor 3)

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Jika \(w = t^{2} - t \: \; tan \, s \) , \(t = x\) dan  \(s = \pi x\), tentukan \(\frac{dw}{dx}\) ketika \(x= \frac{1}{4}\) 

 

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$w=t^2-t\tan s,\ t=x,\ s=\pi x$. Maka $w=x^2-x\tan(\pi x)$. \begin{align*}\frac{dw}{dx}&=2x-\tan(\pi x)-\pi x\sec^2(\pi x)\\\left[\frac{dw}{dx}\right]_{x=\frac{1}{4}}&=\frac{1}{2}-\tan\frac{\pi}{4}-\frac{\pi}{4}\sec^{2}\frac{\pi}{4}\\&=\frac{1}{2}-1-\frac{\pi}{4}\cdot 2\\&=\boxed{-\frac{1}{2}-\frac{\pi}{2}}.\end{align*}

 

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