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Derivative of e^x/x
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Derivative of y=x²+2x+1
Identical expressions
y=sec^ four (x^ three)
y equally sec to the power of 4(x cubed )
y equally sec to the power of four (x to the power of three)
y=sec4(x3)
y=sec4x3
y=sec⁴(x³)
y=sec to the power of 4(x to the power of 3)
y=sec^4x^3

The derivative of the function/ y=sec^4(x^3)

Derivative of y=sec^4(x^3)

The solution

You have entered [src]

   4/ 3\
sec \x /

$$\sec^{4}{\left(x^{3} \right)}$$

d /   4/ 3\\
--\sec \x //
dx

$$\frac{d}{d x} \sec^{4}{\left(x^{3} \right)}$$

Transform

Detail solution

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

$\frac{12 x^{2} \sin{\left(x^{3} \right)} \sec^{3}{\left(x^{3} \right)}}{\cos^{2}{\left(x^{3} \right)}}$
Now simplify:

$\frac{12 x^{2} \sin{\left(x^{3} \right)}}{\cos^{5}{\left(x^{3} \right)}}$

The answer is:

$\frac{12 x^{2} \sin{\left(x^{3} \right)}}{\cos^{5}{\left(x^{3} \right)}}$

The first derivative [src]

    2    4/ 3\    / 3\
12*x *sec \x /*tan\x /

$$12 x^{2} \tan{\left(x^{3} \right)} \sec^{4}{\left(x^{3} \right)}$$

Simplify

The second derivative [src]

        4/ 3\ /     / 3\      3 /       2/ 3\\       3    2/ 3\\
12*x*sec \x /*\2*tan\x / + 3*x *\1 + tan \x // + 12*x *tan \x //

$$12 x \left(3 x^{3} \left(\tan^{2}{\left(x^{3} \right)} + 1\right) + 12 x^{3} \tan^{2}{\left(x^{3} \right)} + 2 \tan{\left(x^{3} \right)}\right) \sec^{4}{\left(x^{3} \right)}$$

Simplify

The third derivative [src]

      4/ 3\ /   3 /       2/ 3\\       3    2/ 3\       6    3/ 3\       6 /       2/ 3\\    / 3\      / 3\\
24*sec \x /*\9*x *\1 + tan \x // + 36*x *tan \x / + 72*x *tan \x / + 63*x *\1 + tan \x //*tan\x / + tan\x //

$$24 \cdot \left(63 x^{6} \left(\tan^{2}{\left(x^{3} \right)} + 1\right) \tan{\left(x^{3} \right)} + 72 x^{6} \tan^{3}{\left(x^{3} \right)} + 9 x^{3} \left(\tan^{2}{\left(x^{3} \right)} + 1\right) + 36 x^{3} \tan^{2}{\left(x^{3} \right)} + \tan{\left(x^{3} \right)}\right) \sec^{4}{\left(x^{3} \right)}$$

Simplify

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Derivative of y=sec^4(x^3)

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