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The derivative of the function/ y=log5x+3x^4

Derivative of y=log5x+3x^4

The solution

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              4
log(5*x) + 3*x

3 x^{4} + \log{\left(5 x \right)}

d /              4\
--\log(5*x) + 3*x /
dx

\frac{d}{d x} \left(3 x^{4} + \log{\left(5 x \right)}\right)

Transform

Detail solution

Differentiate $3 x^{4} + \log{\left(5 x \right)}$ term by term:
1. Let $u = 5 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
  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:
  
  $\frac{1}{x}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
  So, the result is: $12 x^{3}$
The result is: $12 x^{3} + \frac{1}{x}$
Now simplify:

$\frac{12 x^{4} + 1}{x}$

The answer is:

\frac{12 x^{4} + 1}{x}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1       3
- + 12*x 
x

12 x^{3} + \frac{1}{x}

Simplify

The second derivative [src]

  1        2
- -- + 36*x 
   2        
  x

36 x^{2} - \frac{1}{x^{2}}

Simplify

The third derivative [src]

  /1        \
2*|-- + 36*x|
  | 3       |
  \x        /

2 \cdot \left(36 x + \frac{1}{x^{3}}\right)

Simplify

The graph

Derivative of y=log5x+3x^4