Mister Exam

# Derivative of y=x^(3/8)*ctg5x-4x^(-5)

Function f() - derivative -N order at the point
v

from to

### The solution

You have entered [src]
 3/8            4
x   *cot(5*x) - --
5
x 
$$x^{\frac{3}{8}} \cot{\left(5 x \right)} - \frac{4}{x^{5}}$$
d / 3/8            4 \
--|x   *cot(5*x) - --|
dx|                 5|
\                x /
$$\frac{d}{d x} \left(x^{\frac{3}{8}} \cot{\left(5 x \right)} - \frac{4}{x^{5}}\right)$$
Detail solution
1. Differentiate term by term:

1. Apply the product rule:

; to find :

1. Apply the power rule: goes to

; to find :

1. There are multiple ways to do this derivative.

## Method #1

1. Rewrite the function to be differentiated:

2. Let .

3. Apply the power rule: goes to

4. Then, apply the chain rule. Multiply by :

1. Rewrite the function to be differentiated:

2. Apply the quotient rule, which is:

and .

To find :

1. Let .

2. The derivative of sine is cosine:

3. Then, apply the chain rule. Multiply by :

1. The derivative of a constant times a function is the constant times the derivative of the function.

1. Apply the power rule: goes to

So, the result is:

The result of the chain rule is:

To find :

1. Let .

2. The derivative of cosine is negative sine:

3. Then, apply the chain rule. Multiply by :

1. The derivative of a constant times a function is the constant times the derivative of the function.

1. Apply the power rule: goes to

So, the result is:

The result of the chain rule is:

Now plug in to the quotient rule:

The result of the chain rule is:

## Method #2

1. Rewrite the function to be differentiated:

2. Apply the quotient rule, which is:

and .

To find :

1. Let .

2. The derivative of cosine is negative sine:

3. Then, apply the chain rule. Multiply by :

1. The derivative of a constant times a function is the constant times the derivative of the function.

1. Apply the power rule: goes to

So, the result is:

The result of the chain rule is:

To find :

1. Let .

2. The derivative of sine is cosine:

3. Then, apply the chain rule. Multiply by :

1. The derivative of a constant times a function is the constant times the derivative of the function.

1. Apply the power rule: goes to

So, the result is:

The result of the chain rule is:

Now plug in to the quotient rule:

The result is:

2. The derivative of a constant times a function is the constant times the derivative of the function.

1. The derivative of a constant times a function is the constant times the derivative of the function.

1. Apply the power rule: goes to

So, the result is:

So, the result is:

The result is:

2. Now simplify:

The graph
The first derivative [src]
20    3/8 /          2     \   3*cot(5*x)
-- + x   *\-5 - 5*cot (5*x)/ + ----------
6                                  5/8
x                                8*x     
$$x^{\frac{3}{8}} \left(- 5 \cot^{2}{\left(5 x \right)} - 5\right) + \frac{20}{x^{6}} + \frac{3 \cot{\left(5 x \right)}}{8 x^{\frac{5}{8}}}$$
The second derivative [src]
  /         /       2     \                                                \
|  24   3*\1 + cot (5*x)/   3*cot(5*x)       3/8 /       2     \         |
5*|- -- - ----------------- - ---------- + 10*x   *\1 + cot (5*x)/*cot(5*x)|
|   7            5/8             13/8                                    |
\  x          4*x            64*x                                        /
$$5 \cdot \left(10 x^{\frac{3}{8}} \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} - \frac{24}{x^{7}} - \frac{3 \left(\cot^{2}{\left(5 x \right)} + 1\right)}{4 x^{\frac{5}{8}}} - \frac{3 \cot{\left(5 x \right)}}{64 x^{\frac{13}{8}}}\right)$$
The third derivative [src]
  /                             2                    /       2     \                                           /       2     \         \
|168       3/8 /       2     \    39*cot(5*x)   45*\1 + cot (5*x)/        3/8    2      /       2     \   45*\1 + cot (5*x)/*cot(5*x)|
5*|--- - 50*x   *\1 + cot (5*x)/  + ----------- + ------------------ - 100*x   *cot (5*x)*\1 + cot (5*x)/ + ---------------------------|
|  8                                    21/8             13/8                                                           5/8          |
\ x                                512*x             64*x                                                            4*x             /
$$5 \left(- 50 x^{\frac{3}{8}} \left(\cot^{2}{\left(5 x \right)} + 1\right)^{2} - 100 x^{\frac{3}{8}} \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot^{2}{\left(5 x \right)} + \frac{168}{x^{8}} + \frac{45 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)}}{4 x^{\frac{5}{8}}} + \frac{45 \left(\cot^{2}{\left(5 x \right)} + 1\right)}{64 x^{\frac{13}{8}}} + \frac{39 \cot{\left(5 x \right)}}{512 x^{\frac{21}{8}}}\right)$$
The graph