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Identical expressions
с^(sin(a*x+b))
с to the power of ( sinus of (a multiply by x plus b))
с(sin(a*x+b))
сsina*x+b
с^(sin(ax+b))
с(sin(ax+b))
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Similar expressions
с^(sin(a*x-b))

The derivative of the function/ с^(sin(a*x+b))

Derivative of с^(sin(a*x+b))

The solution

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 sin(a*x + b)
c

$$c^{\sin{\left(a x + b \right)}}$$

c^sin(a*x + b)

Detail solution

Let $u = \sin{\left(a x + b \right)}$ .
$\frac{\partial}{\partial u} c^{u} = c^{u} \log{\left(c \right)}$
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \sin{\left(a x + b \right)}$ :
1. Let $u = a x + b$ .
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 $\frac{\partial}{\partial x} \left(a x + b\right)$ :
  1. Differentiate $a x + b$ 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: $a$
    2. The derivative of the constant $b$ is zero.
    The result is: $a$
  The result of the chain rule is:
  
  $a \cos{\left(a x + b \right)}$
The result of the chain rule is:

$a c^{\sin{\left(a x + b \right)}} \log{\left(c \right)} \cos{\left(a x + b \right)}$
Now simplify:

$a c^{\sin{\left(a x + b \right)}} \log{\left(c \right)} \cos{\left(a x + b \right)}$

The answer is:

$a c^{\sin{\left(a x + b \right)}} \log{\left(c \right)} \cos{\left(a x + b \right)}$

The first derivative [src]

   sin(a*x + b)                    
a*c            *cos(a*x + b)*log(c)

$$a c^{\sin{\left(a x + b \right)}} \log{\left(c \right)} \cos{\left(a x + b \right)}$$

Simplify

The second derivative [src]

 2  sin(b + a*x) /                   2                \       
a *c            *\-sin(b + a*x) + cos (b + a*x)*log(c)/*log(c)

$$a^{2} c^{\sin{\left(a x + b \right)}} \left(\log{\left(c \right)} \cos^{2}{\left(a x + b \right)} - \sin{\left(a x + b \right)}\right) \log{\left(c \right)}$$

Simplify

The third derivative [src]

 3  sin(b + a*x) /        2             2                           \                    
a *c            *\-1 + cos (b + a*x)*log (c) - 3*log(c)*sin(b + a*x)/*cos(b + a*x)*log(c)

$$a^{3} c^{\sin{\left(a x + b \right)}} \left(\log{\left(c \right)}^{2} \cos^{2}{\left(a x + b \right)} - 3 \log{\left(c \right)} \sin{\left(a x + b \right)} - 1\right) \log{\left(c \right)} \cos{\left(a x + b \right)}$$

Simplify

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Derivative of с^(sin(a*x+b))

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