Cálculo Ejemplos

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

Paso 1

Diferencia con la regla de la cadena, que establece que $\frac{d}{d x} [f (g (x))]$ es $f' (g (x)) g' (x)$ donde $f (x) = \tan (x)$ y $g (x) = \sin (x^{2})$ .

Toca para ver más pasos...

Paso 1.1

Para aplicar la regla de la cadena, establece $u_{1}$ como $\sin (x^{2})$ .

$\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [\sin (x^{2})]$

Paso 1.2

La derivada de $\tan (u_{1})$ con respecto a $u_{1}$ es $\sec^{2} (u_{1})$ .

$\sec^{2} (u_{1}) \frac{d}{d x} [\sin (x^{2})]$

Paso 1.3

Reemplaza todos los casos de $u_{1}$ con $\sin (x^{2})$ .

$\sec^{2} (\sin (x^{2})) \frac{d}{d x} [\sin (x^{2})]$

Paso 2

Diferencia con la regla de la cadena, que establece que $\frac{d}{d x} [f (g (x))]$ es $f' (g (x)) g' (x)$ donde $f (x) = \sin (x)$ y $g (x) = x^{2}$ .

Toca para ver más pasos...

Paso 2.1

Para aplicar la regla de la cadena, establece $u_{2}$ como $x^{2}$ .

$\sec^{2} (\sin (x^{2})) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [x^{2}])$

Paso 2.2

La derivada de $\sin (u_{2})$ con respecto a $u_{2}$ es $\cos (u_{2})$ .

$\sec^{2} (\sin (x^{2})) (\cos (u_{2}) \frac{d}{d x} [x^{2}])$

Paso 2.3

Reemplaza todos los casos de $u_{2}$ con $x^{2}$ .

$\sec^{2} (\sin (x^{2})) (\cos (x^{2}) \frac{d}{d x} [x^{2}])$

Paso 3

Diferencia con la regla de la potencia, que establece que $\frac{d}{d x} [x^{n}]$ es $n x^{n - 1}$ donde $n = 2$ .

$\sec^{2} (\sin (x^{2})) \cos (x^{2}) (2 x)$

Paso 4

Simplifica.

Toca para ver más pasos...

Paso 4.1

Reordena los factores de $\sec^{2} (\sin (x^{2})) \cos (x^{2}) (2 x)$ .

$2 x \sec^{2} (\sin (x^{2})) \cos (x^{2})$

Paso 4.2

Reescribe $\sec (\sin (x^{2}))$ en términos de senos y cosenos.

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

Paso 4.3

Aplica la regla del producto a $\frac{1}{\cos (\sin (x^{2}))}$ .

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

Paso 4.4

Uno elevado a cualquier potencia es uno.

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

Paso 4.5

Multiplica $2 x \frac{1}{\cos^{2} (\sin (x^{2}))}$ .

Toca para ver más pasos...

Paso 4.5.1

Combina $2$ y $\frac{1}{\cos^{2} (\sin (x^{2}))}$ .

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

Paso 4.5.2

Combina $x$ y $\frac{2}{\cos^{2} (\sin (x^{2}))}$ .

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

Paso 4.6

Mueve $2$ a la izquierda de $x$ .

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

Paso 4.7

Combina $\frac{2 x}{\cos^{2} (\sin (x^{2}))}$ y $\cos (x^{2})$ .

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

Paso 4.8

Factoriza $\cos (\sin (x^{2}))$ de $\cos^{2} (\sin (x^{2}))$ .

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

Paso 4.9

Separa las fracciones.

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

Paso 4.10

Reescribe $\frac{\cos (x^{2})}{\cos (\sin (x^{2}))}$ como un producto.

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

Paso 4.11

Escribe $\cos (x^{2})$ como una fracción con el denominador $1$ .

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

Paso 4.12

Simplifica.

Toca para ver más pasos...

Paso 4.12.1

Divide $\cos (x^{2})$ por $1$ .

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

Paso 4.12.2

Convierte de $\frac{1}{\cos (\sin (x^{2}))}$ a $\sec (\sin (x^{2}))$ .

$\frac{2 x}{\cos (\sin (x^{2}))} (\cos (x^{2}) \sec (\sin (x^{2})))$

Paso 4.13

Multiplica $\frac{2 x}{\cos (\sin (x^{2}))} (\cos (x^{2}) \sec (\sin (x^{2})))$ .

Toca para ver más pasos...

Paso 4.13.1

Combina $\cos (x^{2})$ y $\frac{2 x}{\cos (\sin (x^{2}))}$ .

$\frac{\cos (x^{2}) (2 x)}{\cos (\sin (x^{2}))} \sec (\sin (x^{2}))$

Paso 4.13.2

Combina $\frac{\cos (x^{2}) (2 x)}{\cos (\sin (x^{2}))}$ y $\sec (\sin (x^{2}))$ .

$\frac{\cos (x^{2}) (2 x) \sec (\sin (x^{2}))}{\cos (\sin (x^{2}))}$

Paso 4.14

Separa las fracciones.

$\frac{(2 x) \sec (\sin (x^{2}))}{1} \cdot \frac{\cos (x^{2})}{\cos (\sin (x^{2}))}$

Paso 4.15

Reescribe $\frac{\cos (x^{2})}{\cos (\sin (x^{2}))}$ como un producto.

$\frac{(2 x) \sec (\sin (x^{2}))}{1} (\cos (x^{2}) \frac{1}{\cos (\sin (x^{2}))})$

Paso 4.16

Escribe $\cos (x^{2})$ como una fracción con el denominador $1$ .

$\frac{(2 x) \sec (\sin (x^{2}))}{1} (\frac{\cos (x^{2})}{1} \cdot \frac{1}{\cos (\sin (x^{2}))})$

Paso 4.17

Simplifica.

Toca para ver más pasos...

Paso 4.17.1

Divide $\cos (x^{2})$ por $1$ .

$\frac{(2 x) \sec (\sin (x^{2}))}{1} (\cos (x^{2}) \frac{1}{\cos (\sin (x^{2}))})$

Paso 4.17.2

Convierte de $\frac{1}{\cos (\sin (x^{2}))}$ a $\sec (\sin (x^{2}))$ .

$\frac{(2 x) \sec (\sin (x^{2}))}{1} (\cos (x^{2}) \sec (\sin (x^{2})))$

Paso 4.18

Divide $(2 x) \sec (\sin (x^{2}))$ por $1$ .

$(2 x) \sec (\sin (x^{2})) (\cos (x^{2}) \sec (\sin (x^{2})))$

Paso 4.19

Multiplica $(2 x) \sec (\sin (x^{2})) (\cos (x^{2}) \sec (\sin (x^{2})))$ .

Toca para ver más pasos...

Paso 4.19.1

Eleva $\sec (\sin (x^{2}))$ a la potencia de $1$ .

$2 x (\sec^{1} (\sin (x^{2})) \sec (\sin (x^{2}))) \cos (x^{2})$

Paso 4.19.2

Eleva $\sec (\sin (x^{2}))$ a la potencia de $1$ .

$2 x (\sec^{1} (\sin (x^{2})) \sec^{1} (\sin (x^{2}))) \cos (x^{2})$

Paso 4.19.3

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$2 x {\sec (\sin (x^{2}))}^{1 + 1} \cos (x^{2})$

Paso 4.19.4

Suma $1$ y $1$ .

$2 x \sec^{2} (\sin (x^{2})) \cos (x^{2})$