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tan(x^2+1)
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tan(x^2-1)

The derivative of the function/ tan(x^2+1)

Derivative of tan(x^2+1)

The solution

You have entered [src]

   / 2    \
tan\x  + 1/

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

tan(x^2 + 1)

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(x^{2} + 1 \right)} = \frac{\sin{\left(x^{2} + 1 \right)}}{\cos{\left(x^{2} + 1 \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(x^{2} + 1 \right)}$ and $g{\left(x \right)} = \cos{\left(x^{2} + 1 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{2} + 1$ .
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 $\frac{d}{d x} \left(x^{2} + 1\right)$ :
  1. Differentiate $x^{2} + 1$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $1$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $2 x \cos{\left(x^{2} + 1 \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{2} + 1$ .
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 $\frac{d}{d x} \left(x^{2} + 1\right)$ :
  1. Differentiate $x^{2} + 1$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $1$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $- 2 x \sin{\left(x^{2} + 1 \right)}$
Now plug in to the quotient rule:

$\frac{2 x \sin^{2}{\left(x^{2} + 1 \right)} + 2 x \cos^{2}{\left(x^{2} + 1 \right)}}{\cos^{2}{\left(x^{2} + 1 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /       2/ 2    \\
2*x*\1 + tan \x  + 1//

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

Simplify

The second derivative [src]

  /       2/     2\      2 /       2/     2\\    /     2\\
2*\1 + tan \1 + x / + 4*x *\1 + tan \1 + x //*tan\1 + x //

$$2 \left(4 x^{2} \left(\tan^{2}{\left(x^{2} + 1 \right)} + 1\right) \tan{\left(x^{2} + 1 \right)} + \tan^{2}{\left(x^{2} + 1 \right)} + 1\right)$$

Simplify

The third derivative [src]

    /       2/     2\\ /     /     2\      2 /       2/     2\\      2    2/     2\\
8*x*\1 + tan \1 + x //*\3*tan\1 + x / + 2*x *\1 + tan \1 + x // + 4*x *tan \1 + x //

$$8 x \left(\tan^{2}{\left(x^{2} + 1 \right)} + 1\right) \left(2 x^{2} \left(\tan^{2}{\left(x^{2} + 1 \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} + 1 \right)} + 3 \tan{\left(x^{2} + 1 \right)}\right)$$

Simplify

The graph

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Derivative of tan(x^2+1)

The graph:

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The solution