|
| |
|
|
|
|
| Derivada das Funções Trigonométricas |
| Derivative of Trigonometric Functions |
| |
| \(\dfrac{d}{dx}\sin(x)=\cos(x)\)
\begin{aligned}
\dfrac{d}{dx}\sin(x) =\,&\, \lim \limits_{\Delta x\to 0} \dfrac{\sin(x+\Delta x)-\sin(x)}{\Delta x} \\
=\,&\, \lim \limits_{\Delta x\to 0} \dfrac{\sin(x)\cos(\Delta x)+\sin(\Delta x)\cos(x)-\sin(x)}{\Delta x} \\
=\,&\, \lim \limits_{\Delta x\to 0} \left[\cos(x)\dfrac{\sin(\Delta x)}{\Delta x} - \sin(x)\dfrac{1-\cos(\Delta x)}{\Delta x} \right] \\
=\,&\, \cos(x)\lim \limits_{\Delta x\to 0}\dfrac{\sin(\Delta x)}{\Delta x} - \sin(x)\lim \limits_{\Delta x\to 0}\dfrac{1-\cos(\Delta x)}{\Delta x} \\
=\,&\, \cos(x)(1) - \sin(x)(0) \\
=\,&\, \cos(x)
\end{aligned}
|
\(\dfrac{d}{dx}\cos(x)=-\sin(x)\)
\begin{aligned}
\dfrac{d}{dx}\cos(x) =\,&\, \lim \limits_{\Delta x\to 0} \dfrac{\cos(x+\Delta x)-\cos(x)}{\Delta x} \\
=\,&\, \lim \limits_{\Delta x\to 0} \dfrac{\cos(x)\cos(\Delta x)-\sin(\Delta x)\sin(x)-\cos(x)}{\Delta x} \\
=\,&\, \lim \limits_{\Delta x\to 0} \left[-\sin(x)\dfrac{\sin(\Delta x)}{\Delta x} - \cos(x)\dfrac{1-\cos(\Delta x)}{\Delta x} \right] \\
=\,&\, -\sin(x)\lim \limits_{\Delta x\to 0}\dfrac{\sin(\Delta x)}{\Delta x} - \cos(x)\lim \limits_{\Delta x\to 0}\dfrac{1-\cos(\Delta x)}{\Delta x} \\
=\,&\, -\sin(x)(1) - \cos(x)(0) \\
=\,&\, -\sin(x)
\end{aligned}
|
| \(\dfrac{d}{dx}\tan(x)=\sec^2(x)\)
\begin{aligned}
\dfrac{d}{dx}\tan(x) =\,&\, \dfrac{d}{dx}\left[\dfrac{\sin(x)}{\cos(x)}\right] \\
=\,&\, \dfrac{\cos(x)\dfrac{d}{dx}\sin(x)-\sin(x)\dfrac{d}{dx}\cos(x)}{\cos^2(x)} \\
=\,&\, \dfrac{\cos(x)\cos(x)+\sin(x)\sin(x)}{\cos^2(x)} \\
=\,&\, \dfrac{\cos^2(x)+\sin^2(x)}{\cos^2(x)} \\
=\,&\, \dfrac{1}{\cos^2(x)} = \sec^2(x)
\end{aligned}
|
\(\dfrac{d}{dx}\cot(x)=-\csc^2(x)\)
\begin{aligned}
\dfrac{d}{dx}\cot(x) =\,&\, \dfrac{d}{dx}\left[\dfrac{\cos(x)}{\sin(x)}\right] \\
=\,&\, \dfrac{\sin(x)\dfrac{d}{dx}\cos(x)-\cos(x)\dfrac{d}{dx}\sin(x)}{\sin^2(x)} \\
=\,&\, \dfrac{-\sin(x)\sin(x)-\cos(x)\cos(x)}{\sin^2(x)} \\
=\,&\, \dfrac{-\left[\sin^2(x)+\cos^2(x)\right]}{\sin^2(x)} \\
=\,&\, \dfrac{-1}{\sin^2(x)} = -\csc^2(x)
\end{aligned}
|
| \(\dfrac{d}{dx}\sec(x)=\sec(x)\tan(x)\)
\begin{aligned}
\dfrac{d}{dx}\sec(x) =\,&\, \dfrac{d}{dx}\left[\dfrac{1}{\cos(x)}\right] \\
=\,&\, \dfrac{\cos(x)\dfrac{d}{dx}(1)-(1)\dfrac{d}{dx}\cos(x)}{\cos^2(x)} \\
=\,&\, \dfrac{\cos(x)(0)+(1)\sin(x)}{\cos^2(x)} \\
=\,&\, \dfrac{\sin(x)}{\cos^2(x)} = \dfrac{1}{\cos(x)}\dfrac{\sin(x)}{\cos(x)}\\
=\,&\, \sec(x)\tan(x)
\end{aligned}
|
\(\dfrac{d}{dx}\csc(x)=-\csc(x)\cot(x)\)
\begin{aligned}
\dfrac{d}{dx}\csc(x) =\,&\, \dfrac{d}{dx}\left[\dfrac{1}{\sin(x)}\right] \\
=\,&\, \dfrac{\sin(x)\dfrac{d}{dx}(1)-(1)\dfrac{d}{dx}\sin(x)}{\sin^2(x)} \\
=\,&\, \dfrac{\sin(x)(0)-(1)\cos(x)}{\sin^2(x)} \\
=\,&\, \dfrac{-\cos(x)}{\sin^2(x)} = \dfrac{-1}{\sin(x)}\dfrac{\cos(x)}{\sin(x)}\\
=\,&\, -\csc(x)\cot(x)
\end{aligned}
|
| |
|
|
|
|
|
|
|
