derivada de (tan^2(x)/(tan(x^2)))

1) Vamos derivar:

\frac{d}{d x} (\frac{{\tan (x)}^{2}}{\tan (x^{2})})

Para

p1

{\tan (x)}^{2}

q2

\tan (x^{2})

, utilizamos a fórmula de derivação:

\frac{d}{d x} (\frac{p1}{q2}) = \frac{\frac{d}{d x} (p1)}{q2} - p1 \cdot \frac{\frac{d}{d x} (q2)}{{q2}^{2}}

2) Vamos derivar

p1

\frac{d}{d x} ({\tan (x)}^{2})

Utilizamos a fórmula de derivação

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

com f =

\tan (x)

ⓘ A derivada de

x^{n}

n x^{n - 1}

\frac{d}{d x} ({\tan (x)}^{2})

2 \cdot \frac{d}{d x} (\tan (x)) \cdot \tan (x)

2 \cdot \frac{d}{d x} (\tan (x)) \cdot \tan (x)

ⓘ A derivada de

\tan (x)

\frac{1}{{\cos (x)}^{2}}

\frac{d}{d x} (\tan (x))

\frac{1}{{\cos (x)}^{2}}

2 \cdot \frac{1}{{\cos (x)}^{2}} \cdot \tan (x)

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}

3) Vamos derivar

q2

\frac{d}{d x} (\tan (x^{2}))

Aplicamos a fórmula de derivação

\frac{d}{d x} (\tan (f)) = \frac{d}{d x} (f) \cdot \frac{1}{{\cos (f)}^{2}}

com f =

x^{2}

ⓘ A derivada de

\tan (x)

\frac{1}{{\cos (x)}^{2}}

\frac{d}{d x} (\tan (x^{2}))

\frac{d}{d x} (x^{2}) \cdot \frac{1}{{\cos (x^{2})}^{2}}

\frac{d}{d x} (x^{2}) \cdot \frac{1}{{\cos (x^{2})}^{2}}

ⓘ A derivada de

x^{n}

n x^{n - 1}

\frac{d}{d x} (x^{2})

2 x

\frac{2 x}{{\cos (x^{2})}^{2}}

4) Avaliamos a derivada obtida (

\frac{p1}{q2}

)':

\frac{\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}}{\tan (x^{2})} - \frac{{\tan (x)}^{2} \cdot \frac{2 x}{{\cos (x^{2})}^{2}}}{{\tan (x^{2})}^{2}}

ⓘ Multiplicar uma fração por um termo equivale a multiplicar o seu numerador:

{\tan (x)}^{2} \cdot \frac{2 x}{{\cos (x^{2})}^{2}}

\frac{{\tan (x)}^{2} \cdot 2 x}{{\cos (x^{2})}^{2}}

\frac{\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}}{\tan (x^{2})} - \frac{\frac{{\tan (x)}^{2} \cdot 2 x}{{\cos (x^{2})}^{2}}}{{\tan (x^{2})}^{2}}

ⓘ Dividir uma fração por

\tan (x^{2})

equivale a multiplicar o seu denominador por

\tan (x^{2})

\frac{\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}}{\tan (x^{2})}

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2} \cdot \tan (x^{2})}

■Dividir uma fração por

{\tan (x^{2})}^{2}

equivale a multiplicar o seu denominador por

{\tan (x^{2})}^{2}

- \frac{\frac{{\tan (x)}^{2} \cdot 2 x}{{\cos (x^{2})}^{2}}}{{\tan (x^{2})}^{2}}

- \frac{{\tan (x)}^{2} \cdot 2 x}{{\cos (x^{2})}^{2} \cdot {\tan (x^{2})}^{2}}

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2} \cdot \tan (x^{2})} - \frac{{\tan (x)}^{2} \cdot 2 x}{{\cos (x^{2})}^{2} \cdot {\tan (x^{2})}^{2}}

ⓘ Quando

\cos (x) \neq 0

\tan (x) = \frac{\sin (x)}{\cos (x)}

, pelo que

\tan (x) \cdot \cos (x) = \sin (x)

{\cos (x^{2})}^{2} \cdot {\tan (x^{2})}^{2}

{\sin (x^{2})}^{2}

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2} \cdot \tan (x^{2})} - \frac{{\tan (x)}^{2} \cdot 2 x}{{\sin (x^{2})}^{2}}

5) Depois continuamos a derivação:

ⅰ

\frac{d}{d x} (\frac{{\tan (x)}^{2}}{\tan (x^{2})})

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2} \cdot \tan (x^{2})} - \frac{{\tan (x)}^{2} \cdot 2 x}{{\sin (x^{2})}^{2}}

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2} \cdot \tan (x^{2})} - \frac{{\tan (x)}^{2} \cdot 2 x}{{\sin (x^{2})}^{2}}

▶Derivada:

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2} \cdot \tan (x^{2})} - \frac{2 x \cdot {\tan (x)}^{2}}{{\sin (x^{2})}^{2}}