Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of y=2^x
Derivative of (x^2+2)/(x-3)
Derivative of tan(4*x)+2/3*tan(4*x)^(3)+1/5*tan(4*x)^(5)
Derivative of e^(sin5x)*cosx
Identical expressions
sin(3x)*sqrt(one +x)
sinus of (3x) multiply by square root of (1 plus x)
sinus of (3x) multiply by square root of (one plus x)
sin(3x)*√(1+x)
sin(3x)sqrt(1+x)
sin3xsqrt1+x
Similar expressions
sin(3x)*sqrt(1-x)

The derivative of the function/ sin(3x)*sqrt(1+x)

Derivative of sin(3x)*sqrt(1+x)

The solution

You have entered [src]

           _______
sin(3*x)*\/ 1 + x

$$\sqrt{x + 1} \sin{\left(3 x \right)}$$

sin(3*x)*sqrt(1 + x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

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

$\frac{\left(6 x + 6\right) \cos{\left(3 x \right)} + \sin{\left(3 x \right)}}{2 \sqrt{x + 1}}$

The answer is:

$\frac{\left(6 x + 6\right) \cos{\left(3 x \right)} + \sin{\left(3 x \right)}}{2 \sqrt{x + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  sin(3*x)        _______         
----------- + 3*\/ 1 + x *cos(3*x)
    _______                       
2*\/ 1 + x

$$3 \sqrt{x + 1} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{2 \sqrt{x + 1}}$$

Simplify

The second derivative [src]

      _______            3*cos(3*x)     sin(3*x)  
- 9*\/ 1 + x *sin(3*x) + ---------- - ------------
                           _______             3/2
                         \/ 1 + x     4*(1 + x)

$$- 9 \sqrt{x + 1} \sin{\left(3 x \right)} + \frac{3 \cos{\left(3 x \right)}}{\sqrt{x + 1}} - \frac{\sin{\left(3 x \right)}}{4 \left(x + 1\right)^{\frac{3}{2}}}$$

Simplify

4-я производная [src]

  /  18*cos(3*x)        _______              5*sin(3*x)     3*cos(3*x)     9*sin(3*x) \
3*|- ----------- + 27*\/ 1 + x *sin(3*x) - ------------- + ------------ + ------------|
  |     _______                                      7/2            5/2            3/2|
  \   \/ 1 + x                             16*(1 + x)      2*(1 + x)      2*(1 + x)   /

$$3 \left(27 \sqrt{x + 1} \sin{\left(3 x \right)} - \frac{18 \cos{\left(3 x \right)}}{\sqrt{x + 1}} + \frac{9 \sin{\left(3 x \right)}}{2 \left(x + 1\right)^{\frac{3}{2}}} + \frac{3 \cos{\left(3 x \right)}}{2 \left(x + 1\right)^{\frac{5}{2}}} - \frac{5 \sin{\left(3 x \right)}}{16 \left(x + 1\right)^{\frac{7}{2}}}\right)$$

Simplify

The third derivative [src]

  /      _______             9*sin(3*x)    3*cos(3*x)      sin(3*x)  \
3*|- 9*\/ 1 + x *cos(3*x) - ----------- - ------------ + ------------|
  |                             _______            3/2            5/2|
  \                         2*\/ 1 + x    4*(1 + x)      8*(1 + x)   /

$$3 \left(- 9 \sqrt{x + 1} \cos{\left(3 x \right)} - \frac{9 \sin{\left(3 x \right)}}{2 \sqrt{x + 1}} - \frac{3 \cos{\left(3 x \right)}}{4 \left(x + 1\right)^{\frac{3}{2}}} + \frac{\sin{\left(3 x \right)}}{8 \left(x + 1\right)^{\frac{5}{2}}}\right)$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sin(3x)*sqrt(1+x)

The graph:

Piecewise:

The solution