James Stewart - Resolução Seção 3.3 (7ª Edição)

Pg. 178. 1-16 Derive.

1. $f(x)=3x^2-2\cos x \Rightarrow f'(x)=6x+2\sin x$

2. $f(x) = \sqrt{x}\sin{x} \Rightarrow f'(x)=\sqrt{x}\cos x+\sin x\dfrac{1}{2\sqrt{x}}=\dfrac{2x\cos (x+\sin x)}{2 \sqrt{x}}$

3. $f(x) = \sin x + \frac{1}{2} \cot{x} \Rightarrow f'(x)= \cos x -\frac{1}{2} \csc^2x$

4.  $y = 2\sec{x} - \csc{x} \Rightarrow \dfrac{dy}{dx}=2\sec{x}\tan{x} + \csc{x} \cdot \cot{x} $

5. $g(t)=t^3\cos{t}\Rightarrow g'(t)=-t^3\sin{t}+3t^2\cos{t}=3t^2\cos{t}-t^3\sin{t}$

6. $g(t)=4\sec{t}+\tan{t}\Rightarrow g'(t)=4\sec{t}\tan{t}+\sec^2t$

7. $h(\theta)=\csc{\theta}+e^{\theta}\cot{\theta}\Rightarrow h'(\theta)=-\csc{\theta}\cot{\theta}-e^{\theta}\csc^2\theta+e^{\theta}\cot{\theta}$

$=-\csc{\theta}\cot{\theta}+e^{\theta}(\cot{\theta}-\csc^2\theta)$

8. $y=e^u(\cos{u}+cu)\Rightarrow \dfrac{dy}{du}=e^u(-\sin{u}+u)+(\cos{u}+cu)e^u=e^u(\cos{u}-\sin{u}+cu)$

9. $y=\dfrac{x}{2-\tan{x}}\Rightarrow \dfrac{dy}{dx}=\dfrac{(2-tan{x})\cdot 1+x(\sec^2x)}{(2-\tan{x})^2}=\dfrac{2-\tan{x}+x\sec^2x}{(2-\tan{x})^2}$

10. $y=\sin{\theta}\cos{\theta}\Rightarrow \dfrac{dy}{d\theta}=\sin{\theta}(-\sin{\theta}+\cos{\theta}\cos{\theta}=cos^2\theta-\sin^2\theta=\cos^2\theta-(1-\cos^2\theta)=2\cos^2\theta-1$

11. $f(\theta)=\dfrac{\sec{\theta}}{1+\sec{\theta}}$

      $$f'(\theta)=\dfrac{(1+\sec{\theta})\cdot \sec{\theta}\tan{\theta}-\sec{\theta}\cdot \sec{\theta}\tan{\theta}}{(1+\sec{\theta})^2}=\dfrac{\sec{\theta}\tan{\theta}(1+\sec{\theta}-\sec{\theta})}{(1+\sec{\theta})^2}=\dfrac{\sec{\theta}\tan{\theta}}{(1+\sec{\theta})^2}$$