Mister Exam

Lang:

The derivative of the function/ f(x)3sin(x)-8cos(x)

You entered:

f(x)3sin(x)-8cos(x)

What you mean?

f*x^3*sin(x) - 8*cos(x)

Derivative of f(x)3sin(x)-8cos(x)

The solution

You have entered [src]

f*x*3*sin(x) - 8*cos(x)

$$f x 3 \sin{\left(x \right)} - 8 \cos{\left(x \right)}$$

d                          
--(f*x*3*sin(x) - 8*cos(x))
dx

$$\frac{\partial}{\partial x} \left(f x 3 \sin{\left(x \right)} - 8 \cos{\left(x \right)}\right)$$

Transform

Detail solution

Differentiate $f x 3 \sin{\left(x \right)} - 8 \cos{\left(x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
  So, the result is: $3 f \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$
2. 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. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $- 8 \sin{\left(x \right)}$
  So, the result is: $8 \sin{\left(x \right)}$
The result is: $3 f \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) + 8 \sin{\left(x \right)}$

The answer is:

$3 f \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) + 8 \sin{\left(x \right)}$

The first derivative [src]

8*sin(x) + 3*f*sin(x) + 3*f*x*cos(x)

$$3 f x \cos{\left(x \right)} + 3 f \sin{\left(x \right)} + 8 \sin{\left(x \right)}$$

Simplify

The second derivative [src]

8*cos(x) + 6*f*cos(x) - 3*f*x*sin(x)

$$- 3 f x \sin{\left(x \right)} + 6 f \cos{\left(x \right)} + 8 \cos{\left(x \right)}$$

Simplify

The third derivative [src]

-(8*sin(x) + 9*f*sin(x) + 3*f*x*cos(x))

$$- (3 f x \cos{\left(x \right)} + 9 f \sin{\left(x \right)} + 8 \sin{\left(x \right)})$$

Simplify