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Derivative of:
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Derivative of x^2*sin(2*x-3)
Derivative of x^2/cosx
Identical expressions
tg7x*e^(x^ three)
tg7x multiply by e to the power of (x cubed )
tg7x multiply by e to the power of (x to the power of three)
tg7x*e(x3)
tg7x*ex3
tg7x*e^(x³)
tg7x*e to the power of (x to the power of 3)
tg7xe^(x^3)
tg7xe(x3)
tg7xex3
tg7xe^x^3

The derivative of the function/ tg7x*e^(x^3)

Derivative of tg7x*e^(x^3)

The solution

You have entered [src]

          / 3\
          \x /
tan(7*x)*E

$$e^{x^{3}} \tan{\left(7 x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \tan{\left(7 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(7 x \right)} = \frac{\sin{\left(7 x \right)}}{\cos{\left(7 x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(7 x \right)}$ and $g{\left(x \right)} = \cos{\left(7 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 7 x$ .
  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 $\frac{d}{d x} 7 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $7$
    The result of the chain rule is:
    
    $7 \cos{\left(7 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 7 x$ .
  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 $\frac{d}{d x} 7 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $7$
    The result of the chain rule is:
    
    $- 7 \sin{\left(7 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}}{\cos^{2}{\left(7 x \right)}}$
$g{\left(x \right)} = e^{x^{3}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{3}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{3}$ :
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  The result of the chain rule is:
  
  $3 x^{2} e^{x^{3}}$
The result is: $3 x^{2} e^{x^{3}} \tan{\left(7 x \right)} + \frac{\left(7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}\right) e^{x^{3}}}{\cos^{2}{\left(7 x \right)}}$
Now simplify:

$\frac{\left(3 x^{2} \sin{\left(14 x \right)} + 14\right) e^{x^{3}}}{\cos{\left(14 x \right)} + 1}$

The answer is:

$\frac{\left(3 x^{2} \sin{\left(14 x \right)} + 14\right) e^{x^{3}}}{\cos{\left(14 x \right)} + 1}$

The first derivative [src]

                   / 3\         / 3\         
/         2     \  \x /      2  \x /         
\7 + 7*tan (7*x)/*e     + 3*x *e    *tan(7*x)

$$3 x^{2} e^{x^{3}} \tan{\left(7 x \right)} + \left(7 \tan^{2}{\left(7 x \right)} + 7\right) e^{x^{3}}$$

Simplify

The second derivative [src]

                                                                                 / 3\
/    2 /       2     \      /       2     \                /       3\         \  \x /
\42*x *\1 + tan (7*x)/ + 98*\1 + tan (7*x)/*tan(7*x) + 3*x*\2 + 3*x /*tan(7*x)/*e

$$\left(42 x^{2} \left(\tan^{2}{\left(7 x \right)} + 1\right) + 3 x \left(3 x^{3} + 2\right) \tan{\left(7 x \right)} + 98 \left(\tan^{2}{\left(7 x \right)} + 1\right) \tan{\left(7 x \right)}\right) e^{x^{3}}$$

Simplify

The third derivative [src]

                                                                                                                                             / 3\
/  /       6       3\                /       2     \ /         2     \        /       2     \ /       3\        2 /       2     \         \  \x /
\3*\2 + 9*x  + 18*x /*tan(7*x) + 686*\1 + tan (7*x)/*\1 + 3*tan (7*x)/ + 63*x*\1 + tan (7*x)/*\2 + 3*x / + 882*x *\1 + tan (7*x)/*tan(7*x)/*e

$$\left(882 x^{2} \left(\tan^{2}{\left(7 x \right)} + 1\right) \tan{\left(7 x \right)} + 63 x \left(3 x^{3} + 2\right) \left(\tan^{2}{\left(7 x \right)} + 1\right) + 686 \left(\tan^{2}{\left(7 x \right)} + 1\right) \left(3 \tan^{2}{\left(7 x \right)} + 1\right) + 3 \left(9 x^{6} + 18 x^{3} + 2\right) \tan{\left(7 x \right)}\right) e^{x^{3}}$$

Simplify

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Derivative of tg7x*e^(x^3)

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The solution