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tan(5x^ seven +3x^ four)
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tangent of (5x to the power of seven plus 3x to the power of four)
tan(5x7+3x4)
tan5x7+3x4
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tan5x^7+3x^4
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tan(5x^7-3x^4)

The derivative of the function/ tan(5x^7+3x^4)

Derivative of tan(5x^7+3x^4)

The solution

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   /   7      4\
tan\5*x  + 3*x /

$$\tan{\left(5 x^{7} + 3 x^{4} \right)}$$

tan(5*x^7 + 3*x^4)

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(5 x^{7} + 3 x^{4} \right)} = \frac{\sin{\left(5 x^{7} + 3 x^{4} \right)}}{\cos{\left(5 x^{7} + 3 x^{4} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(5 x^{7} + 3 x^{4} \right)}$ and $g{\left(x \right)} = \cos{\left(5 x^{7} + 3 x^{4} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 5 x^{7} + 3 x^{4}$ .
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 $\frac{d}{d x} \left(5 x^{7} + 3 x^{4}\right)$ :
  1. Differentiate $5 x^{7} + 3 x^{4}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$
      So, the result is: $35 x^{6}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
      So, the result is: $12 x^{3}$
    The result is: $35 x^{6} + 12 x^{3}$
  The result of the chain rule is:
  
  $\left(35 x^{6} + 12 x^{3}\right) \cos{\left(5 x^{7} + 3 x^{4} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 5 x^{7} + 3 x^{4}$ .
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 $\frac{d}{d x} \left(5 x^{7} + 3 x^{4}\right)$ :
  1. Differentiate $5 x^{7} + 3 x^{4}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$
      So, the result is: $35 x^{6}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
      So, the result is: $12 x^{3}$
    The result is: $35 x^{6} + 12 x^{3}$
  The result of the chain rule is:
  
  $- \left(35 x^{6} + 12 x^{3}\right) \sin{\left(5 x^{7} + 3 x^{4} \right)}$
Now plug in to the quotient rule:

$\frac{\left(35 x^{6} + 12 x^{3}\right) \sin^{2}{\left(5 x^{7} + 3 x^{4} \right)} + \left(35 x^{6} + 12 x^{3}\right) \cos^{2}{\left(5 x^{7} + 3 x^{4} \right)}}{\cos^{2}{\left(5 x^{7} + 3 x^{4} \right)}}$
Now simplify:

$\frac{x^{3} \left(35 x^{3} + 12\right)}{\cos^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)}}$

The answer is:

$\frac{x^{3} \left(35 x^{3} + 12\right)}{\cos^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/       2/   7      4\\ /    3       6\
\1 + tan \5*x  + 3*x //*\12*x  + 35*x /

$$\left(35 x^{6} + 12 x^{3}\right) \left(\tan^{2}{\left(5 x^{7} + 3 x^{4} \right)} + 1\right)$$

Simplify

The second derivative [src]

                               /                             2                   \
   2 /       2/ 4 /       3\\\ |          3    4 /         3\     / 4 /       3\\|
2*x *\1 + tan \x *\3 + 5*x ///*\18 + 105*x  + x *\12 + 35*x / *tan\x *\3 + 5*x ///

$$2 x^{2} \left(\tan^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 1\right) \left(x^{4} \left(35 x^{3} + 12\right)^{2} \tan{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 105 x^{3} + 18\right)$$

Simplify

The third derivative [src]

                              /                             3                                              3                                                                        \
    /       2/ 4 /       3\\\ |          3    8 /         3\  /       2/ 4 /       3\\\      8 /         3\     2/ 4 /       3\\       4 /        3\ /         3\    / 4 /       3\\|
2*x*\1 + tan \x *\3 + 5*x ///*\36 + 525*x  + x *\12 + 35*x / *\1 + tan \x *\3 + 5*x /// + 2*x *\12 + 35*x / *tan \x *\3 + 5*x // + 18*x *\6 + 35*x /*\12 + 35*x /*tan\x *\3 + 5*x ///

$$2 x \left(\tan^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 1\right) \left(x^{8} \left(35 x^{3} + 12\right)^{3} \left(\tan^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 1\right) + 2 x^{8} \left(35 x^{3} + 12\right)^{3} \tan^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 18 x^{4} \left(35 x^{3} + 6\right) \left(35 x^{3} + 12\right) \tan{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 525 x^{3} + 36\right)$$

Simplify

The graph

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Derivative of tan(5x^7+3x^4)

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The solution