(b) Simplify the expression for f(x) by writing it in terms of sin(x) and cos(x), and then find f '(x). f '(x) =
Added by Paul B.
Step 1
To simplify the expression for \( f(x) \) in terms of \( \sin(x) \) and \( \cos(x) \), and then find \( f'(x) \), we will follow these steps: Show more…
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If $f(x)=x^{2} \cos 2 x,$ find $f^{\prime}(x)$ (A) $-2 x \cos 2 x+2 x^{2} \sin 2 x$ (B) $-4 x \sin 2 x$ (C) $2 x \cos 2 x-2 x^{2} \sin 2 x$ (D) $2 x-2 \sin 2 x$
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We will use the following identities to calculate the derivatives of $\sin x$ and $\cos x$$$\begin{aligned} \sin (a+b) &=\sin a \cos b+\sin b \cos a \\\cos (a+b) &=\cos a \cos b-\sin a \sin b\end{aligned}$$ (a) Use the definition of the derivative to show that if $f(x)=\sin x$$$f^{\prime}(x)=\sin x \lim _{h \rightarrow 0} \frac{\cos h-1}{h}+\cos x \lim _{h \rightarrow 0} \frac{\sin h}{h}$$ (b) Estimate the limits in part (a) with your calculator to explain why $f^{\prime}(x)=\cos x$ (c) If $g(x)=\cos x,$ use the definition of the derivative to show that $g^{\prime}(x)=-\sin x$
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