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Identical expressions
tan(7x^ two - five)
tangent of (7x squared minus 5)
tangent of (7x to the power of two minus five)
tan(7x2-5)
tan7x2-5
tan(7x²-5)
tan(7x to the power of 2-5)
tan7x^2-5
Similar expressions
tan(7x^2+5)

The derivative of the function/ tan(7x^2-5)

Derivative of tan(7x^2-5)

The solution

You have entered [src]

   /   2    \
tan\7*x  - 5/

$$\tan{\left(7 x^{2} - 5 \right)}$$

d /   /   2    \\
--\tan\7*x  - 5//
dx

$$\frac{d}{d x} \tan{\left(7 x^{2} - 5 \right)}$$

Transform

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(7 x^{2} - 5 \right)} = \frac{\sin{\left(7 x^{2} - 5 \right)}}{\cos{\left(7 x^{2} - 5 \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(7 x^{2} - 5 \right)}$ and $g{\left(x \right)} = \cos{\left(7 x^{2} - 5 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 7 x^{2} - 5$ .
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 $\frac{d}{d x} \left(7 x^{2} - 5\right)$ :
  1. Differentiate $7 x^{2} - 5$ 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: $x^{2}$ goes to $2 x$
      So, the result is: $14 x$
    2. The derivative of the constant $\left(-1\right) 5$ is zero.
    The result is: $14 x$
  The result of the chain rule is:
  
  $14 x \cos{\left(7 x^{2} - 5 \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 7 x^{2} - 5$ .
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 $\frac{d}{d x} \left(7 x^{2} - 5\right)$ :
  1. Differentiate $7 x^{2} - 5$ 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: $x^{2}$ goes to $2 x$
      So, the result is: $14 x$
    2. The derivative of the constant $\left(-1\right) 5$ is zero.
    The result is: $14 x$
  The result of the chain rule is:
  
  $- 14 x \sin{\left(7 x^{2} - 5 \right)}$
Now plug in to the quotient rule:

$\frac{14 x \sin^{2}{\left(7 x^{2} - 5 \right)} + 14 x \cos^{2}{\left(7 x^{2} - 5 \right)}}{\cos^{2}{\left(7 x^{2} - 5 \right)}}$
Now simplify:

$\frac{14 x}{\cos^{2}{\left(7 x^{2} - 5 \right)}}$

The answer is:

$\frac{14 x}{\cos^{2}{\left(7 x^{2} - 5 \right)}}$

The graph

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The first derivative [src]

     /       2/   2    \\
14*x*\1 + tan \7*x  - 5//

$$14 x \left(\tan^{2}{\left(7 x^{2} - 5 \right)} + 1\right)$$

Simplify

The second derivative [src]

   /       2/        2\       2 /       2/        2\\    /        2\\
14*\1 + tan \-5 + 7*x / + 28*x *\1 + tan \-5 + 7*x //*tan\-5 + 7*x //

$$14 \cdot \left(28 x^{2} \left(\tan^{2}{\left(7 x^{2} - 5 \right)} + 1\right) \tan{\left(7 x^{2} - 5 \right)} + \tan^{2}{\left(7 x^{2} - 5 \right)} + 1\right)$$

Simplify

The third derivative [src]

      /       2/        2\\ /     /        2\       2 /       2/        2\\       2    2/        2\\
392*x*\1 + tan \-5 + 7*x //*\3*tan\-5 + 7*x / + 14*x *\1 + tan \-5 + 7*x // + 28*x *tan \-5 + 7*x //

$$392 x \left(\tan^{2}{\left(7 x^{2} - 5 \right)} + 1\right) \left(14 x^{2} \left(\tan^{2}{\left(7 x^{2} - 5 \right)} + 1\right) + 28 x^{2} \tan^{2}{\left(7 x^{2} - 5 \right)} + 3 \tan{\left(7 x^{2} - 5 \right)}\right)$$

Simplify

The graph

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Derivative of tan(7x^2-5)

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