Derivative of y=sec(3x+5)

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sec(3*x + 5)

$$\sec{\left(3 x + 5 \right)}$$

sec(3*x + 5)

Detail solution

Rewrite the function to be differentiated:

$\sec{\left(3 x + 5 \right)} = \frac{1}{\cos{\left(3 x + 5 \right)}}$
Let $u = \cos{\left(3 x + 5 \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(3 x + 5 \right)}$ :
1. Let $u = 3 x + 5$ .
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 $\frac{d}{d x} \left(3 x + 5\right)$ :
  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:
  
  $- 3 \sin{\left(3 x + 5 \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

3*sec(3*x + 5)*tan(3*x + 5)

$$3 \tan{\left(3 x + 5 \right)} \sec{\left(3 x + 5 \right)}$$

Simplify

The second derivative [src]

  /         2         \             
9*\1 + 2*tan (5 + 3*x)/*sec(5 + 3*x)

$$9 \left(2 \tan^{2}{\left(3 x + 5 \right)} + 1\right) \sec{\left(3 x + 5 \right)}$$

Simplify

The third derivative [src]

   /         2         \                          
27*\5 + 6*tan (5 + 3*x)/*sec(5 + 3*x)*tan(5 + 3*x)

$$27 \left(6 \tan^{2}{\left(3 x + 5 \right)} + 5\right) \tan{\left(3 x + 5 \right)} \sec{\left(3 x + 5 \right)}$$

Simplify

The graph