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Identical expressions
√(3x+ five)+ one /(cos²x)
√(3x plus 5) plus 1 divide by ( co sinus of e of ²x)
√(3x plus five) plus one divide by ( co sinus of e of ²x)
√3x+5+1/cos²x
√(3x+5)+1 divide by (cos²x)
Similar expressions
√(3x-5)+1/(cos²x)
√(3x+5)-1/(cos²x)

The derivative of the function/ √(3x+5)+1/(cos²x)

Derivative of √(3x+5)+1/(cos²x)

The solution

You have entered [src]

  _________      1   
\/ 3*x + 5  + -------
                 2   
              cos (x)

$$\sqrt{3 x + 5} + \frac{1}{\cos^{2}{\left(x \right)}}$$

sqrt(3*x + 5) + 1/(cos(x)^2)

Detail solution

Differentiate $\sqrt{3 x + 5} + \frac{1}{\cos^{2}{\left(x \right)}}$ term by term:
1. Let $u = 3 x + 5$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 5\right)$ :
  1. Differentiate $3 x + 5$ 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: $x$ goes to $1$
      So, the result is: $3$
    2. The derivative of the constant $5$ is zero.
    The result is: $3$
  The result of the chain rule is:
  
  $\frac{3}{2 \sqrt{3 x + 5}}$
4. Let $u = \cos^{2}{\left(x \right)}$ .
5. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
6. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos^{2}{\left(x \right)}$ :
  1. Let $u = \cos{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$
The result is: $\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{3}{2 \sqrt{3 x + 5}}$
Now simplify:

$\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{3}{2 \sqrt{3 x + 5}}$

The answer is:

$\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{3}{2 \sqrt{3 x + 5}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      3            2*sin(x)   
------------- + --------------
    _________             2   
2*\/ 3*x + 5    cos(x)*cos (x)

$$\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)} \cos^{2}{\left(x \right)}} + \frac{3}{2 \sqrt{3 x + 5}}$$

Simplify

The second derivative [src]

                                2   
   2            9          6*sin (x)
------- - -------------- + ---------
   2                 3/2       4    
cos (x)   4*(5 + 3*x)       cos (x)

$$\frac{6 \sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{2}{\cos^{2}{\left(x \right)}} - \frac{9}{4 \left(3 x + 5\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

                                   3   
      81         16*sin(x)   24*sin (x)
-------------- + --------- + ----------
           5/2       3           5     
8*(5 + 3*x)       cos (x)     cos (x)

$$\frac{24 \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + \frac{16 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{81}{8 \left(3 x + 5\right)^{\frac{5}{2}}}$$

Simplify

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Derivative of √(3x+5)+1/(cos²x)

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The solution