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Identical expressions
y=sin(7x- five)*ln(4x+ five)
y equally sinus of (7x minus 5) multiply by ln(4x plus 5)
y equally sinus of (7x minus five) multiply by ln(4x plus five)
y=sin(7x-5)ln(4x+5)
y=sin7x-5ln4x+5
Similar expressions
y=sin(7x-5)*ln(4x-5)
y=sin(7x+5)*ln(4x+5)

The derivative of the function/ y=sin(7x-5)*ln(4x+5)

Derivative of y=sin(7x-5)*ln(4x+5)

The solution

You have entered [src]

sin(7*x - 5)*log(4*x + 5)

$$\log{\left(4 x + 5 \right)} \sin{\left(7 x - 5 \right)}$$

sin(7*x - 5)*log(4*x + 5)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(7 x - 5 \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 7 x - 5$ .
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 $\frac{d}{d x} \left(7 x - 5\right)$ :
  1. Differentiate $7 x - 5$ 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: $7$
    2. The derivative of the constant $-5$ is zero.
    The result is: $7$
  The result of the chain rule is:
  
  $7 \cos{\left(7 x - 5 \right)}$
$g{\left(x \right)} = \log{\left(4 x + 5 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x + 5$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 5\right)$ :
  1. Differentiate $4 x + 5$ 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: $4$
    2. The derivative of the constant $5$ is zero.
    The result is: $4$
  The result of the chain rule is:
  
  $\frac{4}{4 x + 5}$
The result is: $7 \log{\left(4 x + 5 \right)} \cos{\left(7 x - 5 \right)} + \frac{4 \sin{\left(7 x - 5 \right)}}{4 x + 5}$
Now simplify:

$\frac{7 \left(4 x + 5\right) \log{\left(4 x + 5 \right)} \cos{\left(7 x - 5 \right)} + 4 \sin{\left(7 x - 5 \right)}}{4 x + 5}$

The answer is:

$\frac{7 \left(4 x + 5\right) \log{\left(4 x + 5 \right)} \cos{\left(7 x - 5 \right)} + 4 \sin{\left(7 x - 5 \right)}}{4 x + 5}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

4*sin(7*x - 5)                              
-------------- + 7*cos(7*x - 5)*log(4*x + 5)
   4*x + 5

$$7 \log{\left(4 x + 5 \right)} \cos{\left(7 x - 5 \right)} + \frac{4 \sin{\left(7 x - 5 \right)}}{4 x + 5}$$

Simplify

The second derivative [src]

                                 16*sin(-5 + 7*x)   56*cos(-5 + 7*x)
-49*log(5 + 4*x)*sin(-5 + 7*x) - ---------------- + ----------------
                                             2          5 + 4*x     
                                    (5 + 4*x)

$$- 49 \log{\left(4 x + 5 \right)} \sin{\left(7 x - 5 \right)} + \frac{56 \cos{\left(7 x - 5 \right)}}{4 x + 5} - \frac{16 \sin{\left(7 x - 5 \right)}}{\left(4 x + 5\right)^{2}}$$

Simplify

The third derivative [src]

  588*sin(-5 + 7*x)                                    336*cos(-5 + 7*x)   128*sin(-5 + 7*x)
- ----------------- - 343*cos(-5 + 7*x)*log(5 + 4*x) - ----------------- + -----------------
       5 + 4*x                                                      2                   3   
                                                           (5 + 4*x)           (5 + 4*x)

$$- 343 \log{\left(4 x + 5 \right)} \cos{\left(7 x - 5 \right)} - \frac{588 \sin{\left(7 x - 5 \right)}}{4 x + 5} - \frac{336 \cos{\left(7 x - 5 \right)}}{\left(4 x + 5\right)^{2}} + \frac{128 \sin{\left(7 x - 5 \right)}}{\left(4 x + 5\right)^{3}}$$

Simplify

The graph

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Derivative of y=sin(7x-5)*ln(4x+5)

The graph:

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The solution