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The derivative of the function/ sec^3x

Derivative of sec^3x

The solution

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   3   
sec (x)

$$\sec^{3}{\left(x \right)}$$

d /   3   \
--\sec (x)/
dx

$$\frac{d}{d x} \sec^{3}{\left(x \right)}$$

Transform

Detail solution

Let $u = \sec{\left(x \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sec{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
2. Let $u = \cos{\left(x \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. 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:
  
  $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

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

$\frac{3 \sin{\left(x \right)}}{\cos^{4}{\left(x \right)}}$

The answer is:

$\frac{3 \sin{\left(x \right)}}{\cos^{4}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3          
3*sec (x)*tan(x)

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

Simplify

The second derivative [src]

     3    /         2   \
3*sec (x)*\1 + 4*tan (x)/

$$3 \cdot \left(4 \tan^{2}{\left(x \right)} + 1\right) \sec^{3}{\left(x \right)}$$

Simplify

The third derivative [src]

     3    /           2   \       
3*sec (x)*\11 + 20*tan (x)/*tan(x)

$$3 \cdot \left(20 \tan^{2}{\left(x \right)} + 11\right) \tan{\left(x \right)} \sec^{3}{\left(x \right)}$$

Simplify

The graph

Derivative of sec^3x