Derivative of y=tg(sinx)

The solution

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tan(sin(x))

$$\tan{\left(\sin{\left(x \right)} \right)}$$

d              
--(tan(sin(x)))
dx

$$\frac{d}{d x} \tan{\left(\sin{\left(x \right)} \right)}$$

Transform

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\sin{\left(x \right)} \right)} = \frac{\sin{\left(\sin{\left(x \right)} \right)}}{\cos{\left(\sin{\left(x \right)} \right)}}$
Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(\sin{\left(x \right)} \right)}$ and $g{\left(x \right)} = \cos{\left(\sin{\left(x \right)} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $- \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{\sin^{2}{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} + \cos{\left(x \right)} \cos^{2}{\left(\sin{\left(x \right)} \right)}}{\cos^{2}{\left(\sin{\left(x \right)} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/       2        \       
\1 + tan (sin(x))/*cos(x)

$$\left(\tan^{2}{\left(\sin{\left(x \right)} \right)} + 1\right) \cos{\left(x \right)}$$

Simplify

The second derivative [src]

/       2        \ /               2               \
\1 + tan (sin(x))/*\-sin(x) + 2*cos (x)*tan(sin(x))/

$$\left(2 \cos^{2}{\left(x \right)} \tan{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)}\right) \left(\tan^{2}{\left(\sin{\left(x \right)} \right)} + 1\right)$$

Simplify

The third derivative [src]

/       2        \ /                                 2    /       2        \        2       2        \       
\1 + tan (sin(x))/*\-1 - 6*sin(x)*tan(sin(x)) + 2*cos (x)*\1 + tan (sin(x))/ + 4*cos (x)*tan (sin(x))/*cos(x)

$$\left(\tan^{2}{\left(\sin{\left(x \right)} \right)} + 1\right) \left(4 \cos^{2}{\left(x \right)} \tan^{2}{\left(\sin{\left(x \right)} \right)} + 2 \left(\tan^{2}{\left(\sin{\left(x \right)} \right)} + 1\right) \cos^{2}{\left(x \right)} - 6 \sin{\left(x \right)} \tan{\left(\sin{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)}$$

Simplify

The graph

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Derivative of y=tg(sinx)

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