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The derivative of the function/ y=(log6x+sinx)

Derivative of y=(log6x+sinx)

The solution

You have entered [src]

log(6*x) + sin(x)

$$\log{\left(6 x \right)} + \sin{\left(x \right)}$$

log(6*x) + sin(x)

Detail solution

Differentiate $\log{\left(6 x \right)} + \sin{\left(x \right)}$ term by term:
1. Let $u = 6 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 6 x$ :
  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: $6$
  The result of the chain rule is:
  
  $\frac{1}{x}$
4. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
The result is: $\cos{\left(x \right)} + \frac{1}{x}$

The answer is:

$\cos{\left(x \right)} + \frac{1}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1         
- + cos(x)
x

$$\cos{\left(x \right)} + \frac{1}{x}$$

Simplify

The second derivative [src]

 /1          \
-|-- + sin(x)|
 | 2         |
 \x          /

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

Simplify

The third derivative [src]

          2 
-cos(x) + --
           3
          x

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

Simplify

The graph

Derivative of y=(log6x+sinx)