Derivative of sec5x

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sec(5*x)

$$\sec{\left(5 x \right)}$$

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Detail solution

Rewrite the function to be differentiated:

$\sec{\left(5 x \right)} = \frac{1}{\cos{\left(5 x \right)}}$
Let $u = \cos{\left(5 x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(5 x \right)}$ :
1. Let $u = 5 x$ .
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 $\frac{d}{d x} 5 x$ :
  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: $5$
  The result of the chain rule is:
  
  $- 5 \sin{\left(5 x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

5*sec(5*x)*tan(5*x)

$$5 \tan{\left(5 x \right)} \sec{\left(5 x \right)}$$

Simplify

The second derivative [src]

   /         2     \         
25*\1 + 2*tan (5*x)/*sec(5*x)

$$25 \left(2 \tan^{2}{\left(5 x \right)} + 1\right) \sec{\left(5 x \right)}$$

Simplify

The third derivative [src]

    /         2     \                  
125*\5 + 6*tan (5*x)/*sec(5*x)*tan(5*x)

$$125 \left(6 \tan^{2}{\left(5 x \right)} + 5\right) \tan{\left(5 x \right)} \sec{\left(5 x \right)}$$

Simplify

The graph