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The derivative of the function/ y(x)=sqrt(3x+5)

Derivative of y(x)=sqrt(3x+5)

The solution

You have entered [src]

  _________
\/ 3*x + 5

$$\sqrt{3 x + 5}$$

sqrt(3*x + 5)

Detail solution

Let $u = 3 x + 5$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 5\right)$ :
1. Differentiate $3 x + 5$ term by term:
  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: $3$
  2. The derivative of the constant $5$ is zero.
  The result is: $3$
The result of the chain rule is:

$\frac{3}{2 \sqrt{3 x + 5}}$
Now simplify:

$\frac{3}{2 \sqrt{3 x + 5}}$

The answer is:

$\frac{3}{2 \sqrt{3 x + 5}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      3      
-------------
    _________
2*\/ 3*x + 5

$$\frac{3}{2 \sqrt{3 x + 5}}$$

Simplify

The second derivative [src]

     -9       
--------------
           3/2
4*(5 + 3*x)

$$- \frac{9}{4 \left(3 x + 5\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

      81      
--------------
           5/2
8*(5 + 3*x)

$$\frac{81}{8 \left(3 x + 5\right)^{\frac{5}{2}}}$$

Simplify

The graph

Derivative of y(x)=sqrt(3x+5)