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Derivative of:
Derivative of 3*x^5-2*x^2
Derivative of 2^x*sin(x)
Derivative of (1/x)^x
Derivative of x-ln(1+x)
Identical expressions
y=sin5x−ln(x)^- one
y equally sinus of 5x−ln(x) to the power of minus 1
y equally sinus of 5x−ln(x) to the power of minus one
y=sin5x−ln(x)-1
y=sin5x−lnx-1
y=sin5x−lnx^-1
Similar expressions
y=sin5x−ln(x)^+1

The derivative of the function/ y=sin5x−ln(x)^-1

Derivative of y=sin5x−ln(x)^-1

The solution

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             1   
sin(5*x) - ------
           log(x)

$$\sin{\left(5 x \right)} - \frac{1}{\log{\left(x \right)}}$$

sin(5*x) - 1/log(x)

Detail solution

Differentiate $\sin{\left(5 x \right)} - \frac{1}{\log{\left(x \right)}}$ term by term:
1. Let $u = 5 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
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:
  
  $5 \cos{\left(5 x \right)}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $- \frac{1}{x \log{\left(x \right)}^{2}}$
  So, the result is: $\frac{1}{x \log{\left(x \right)}^{2}}$
The result is: $5 \cos{\left(5 x \right)} + \frac{1}{x \log{\left(x \right)}^{2}}$

The answer is:

$5 \cos{\left(5 x \right)} + \frac{1}{x \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 1    
5*cos(5*x) + ---------
                  2   
             x*log (x)

$$5 \cos{\left(5 x \right)} + \frac{1}{x \log{\left(x \right)}^{2}}$$

Simplify

The second derivative [src]

 /                  1            2     \
-|25*sin(5*x) + ---------- + ----------|
 |               2    2       2    3   |
 \              x *log (x)   x *log (x)/

$$- (25 \sin{\left(5 x \right)} + \frac{1}{x^{2} \log{\left(x \right)}^{2}} + \frac{2}{x^{2} \log{\left(x \right)}^{3}})$$

Simplify

The third derivative [src]

                    2            6            6     
-125*cos(5*x) + ---------- + ---------- + ----------
                 3    2       3    4       3    3   
                x *log (x)   x *log (x)   x *log (x)

$$- 125 \cos{\left(5 x \right)} + \frac{2}{x^{3} \log{\left(x \right)}^{2}} + \frac{6}{x^{3} \log{\left(x \right)}^{3}} + \frac{6}{x^{3} \log{\left(x \right)}^{4}}$$

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of y=sin5x−ln(x)^-1

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Piecewise:

The solution