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The derivative of the function/ y=3x*+5sinx-e*

Derivative of y=3x+5sinx-e

The solution

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3*x*5*sin(x) - E

$$5 \cdot 3 x \sin{\left(x \right)} - e$$

((3*x)*5)*sin(x) - E

Detail solution

Differentiate $5 \cdot 3 x \sin{\left(x \right)} - e$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 5 \cdot 3 x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    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: $3$
    So, the result is: $15$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $15 x \cos{\left(x \right)} + 15 \sin{\left(x \right)}$
2. The derivative of the constant $- e$ is zero.
The result is: $15 x \cos{\left(x \right)} + 15 \sin{\left(x \right)}$

The answer is:

$15 x \cos{\left(x \right)} + 15 \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

15*sin(x) + 15*x*cos(x)

$$15 x \cos{\left(x \right)} + 15 \sin{\left(x \right)}$$

Simplify

The second derivative [src]

15*(2*cos(x) - x*sin(x))

$$15 \left(- x \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

-15*(3*sin(x) + x*cos(x))

$$- 15 \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right)$$

Simplify