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Derivative of tanx
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Derivative of tan(3+2x^2)
Identical expressions
tan(three + two x^2)
tangent of (3 plus 2x squared )
tangent of (three plus two x squared )
tan(3+2x2)
tan3+2x2
tan(3+2x²)
tan(3+2x to the power of 2)
tan3+2x^2
Similar expressions
tan(3-2x^2)

The derivative of the function/ tan(3+2x^2)

Derivative of tan(3+2x^2)

The solution

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   /       2\
tan\3 + 2*x /

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

tan(3 + 2*x^2)

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(2 x^{2} + 3 \right)} = \frac{\sin{\left(2 x^{2} + 3 \right)}}{\cos{\left(2 x^{2} + 3 \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(2 x^{2} + 3 \right)}$ and $g{\left(x \right)} = \cos{\left(2 x^{2} + 3 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x^{2} + 3$ .
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 $\frac{d}{d x} \left(2 x^{2} + 3\right)$ :
  1. Differentiate $2 x^{2} + 3$ term by term:
    1. The derivative of the constant $3$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $4 x$
    The result is: $4 x$
  The result of the chain rule is:
  
  $4 x \cos{\left(2 x^{2} + 3 \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x^{2} + 3$ .
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 $\frac{d}{d x} \left(2 x^{2} + 3\right)$ :
  1. Differentiate $2 x^{2} + 3$ term by term:
    1. The derivative of the constant $3$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $4 x$
    The result is: $4 x$
  The result of the chain rule is:
  
  $- 4 x \sin{\left(2 x^{2} + 3 \right)}$
Now plug in to the quotient rule:

$\frac{4 x \sin^{2}{\left(2 x^{2} + 3 \right)} + 4 x \cos^{2}{\left(2 x^{2} + 3 \right)}}{\cos^{2}{\left(2 x^{2} + 3 \right)}}$
Now simplify:

$\frac{4 x}{\cos^{2}{\left(2 x^{2} + 3 \right)}}$

The answer is:

$\frac{4 x}{\cos^{2}{\left(2 x^{2} + 3 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

$$32 x \left(\tan^{2}{\left(2 x^{2} + 3 \right)} + 1\right) \left(4 x^{2} \left(\tan^{2}{\left(2 x^{2} + 3 \right)} + 1\right) + 8 x^{2} \tan^{2}{\left(2 x^{2} + 3 \right)} + 3 \tan{\left(2 x^{2} + 3 \right)}\right)$$

Simplify

The graph

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Derivative of tan(3+2x^2)

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The solution