First Derivative Calculator(Solver) with Steps

Free derivatives calculator(solver) that gets the detailed solution of the first derivative of a function.

[tex]\frac{d}{d x} e^{- 3 x \sin^{2}{\left (x \right )} + 1}[/tex]

Let [tex]u = - 3 x \sin^{2}{\left (x \right )} + 1[/tex].
The derivative of [tex]e^{u}[/tex] is itself.
Then, apply the chain rule. Multiply by [tex]\frac{d}{d x}\left(- 3 x \sin^{2}{\left (x \right )} + 1\right)[/tex]:
1. Differentiate [tex]- 3 x \sin^{2}{\left (x \right )} + 1[/tex] term by term:
  1. The derivative of the constant [tex]1[/tex] is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the product rule:
      
      [tex]\frac{d}{d x}\left(f{\left (x \right )} g{\left (x \right )}\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 )}[/tex]
      
      [tex]f{\left (x \right )} = x[/tex]; to find [tex]\frac{d}{d x} f{\left (x \right )}[/tex]:
      1. Apply the power rule: [tex]x[/tex] goes to [tex]1[/tex]
      [tex]g{\left (x \right )} = \sin^{2}{\left (x \right )}[/tex]; to find [tex]\frac{d}{d x} g{\left (x \right )}[/tex]:
      1. Let [tex]u = \sin{\left (x \right )}[/tex].
      2. Apply the power rule: [tex]u^{2}[/tex] goes to [tex]2 u[/tex]
      3. Then, apply the chain rule. Multiply by [tex]\frac{d}{d x} \sin{\left (x \right )}[/tex]:
        
        The derivative of sine is cosine:
        
        [tex]\frac{d}{d x} \sin{\left (x \right )} = \cos{\left (x \right )}[/tex]
        
        The result of the chain rule is:
        
        [tex]2 \sin{\left (x \right )} \cos{\left (x \right )}[/tex]
      The result is: [tex]2 x \sin{\left (x \right )} \cos{\left (x \right )} + \sin^{2}{\left (x \right )}[/tex]
    So, the result is: [tex]- 6 x \sin{\left (x \right )} \cos{\left (x \right )} - 3 \sin^{2}{\left (x \right )}[/tex]
  The result is: [tex]- 6 x \sin{\left (x \right )} \cos{\left (x \right )} - 3 \sin^{2}{\left (x \right )}[/tex]
The result of the chain rule is:

[tex]\left(- 6 x \sin{\left (x \right )} \cos{\left (x \right )} - 3 \sin^{2}{\left (x \right )}\right) e^{- 3 x \sin^{2}{\left (x \right )} + 1}[/tex]
Now simplify:

[tex]- \frac{3}{2} \left(2 x \sin{\left (2 x \right )} - \cos{\left (2 x \right )} + 1\right) e^{\frac{3 x}{2} \left(\cos{\left (2 x \right )} - 1\right) + 1}[/tex]

The answer is:

[tex]- \frac{3}{2} \left(2 x \sin{\left (2 x \right )} - \cos{\left (2 x \right )} + 1\right) e^{\frac{3 x}{2} \left(\cos{\left (2 x \right )} - 1\right) + 1}[/tex]

Commands:
* is multiplication
oo is $\infty$
pi is $\pi$
x^2 is x²
sqrt(x) is $\sqrt{x}$
sqrt[3](x) is $\sqrt[3]{x}$
(a+b)/(c+d) is $\frac{a+b}{c+d}$

The Most Important Derivatives - Basic Formulas/Rules

$\frac{d}{dx}a=0$ (a is a constant)

$\frac{d}{dx}x=1$

$\frac{d}{dx}x^n=nx^{n-1}$

$\frac{d}{dx}e^x=e^x$

$\frac{d}{dx}\log x=\frac1x$

$\frac{d}{dx}a^x=a^x\log x$

$(f\ g)' = f'g + fg'$ - Product Rule

$(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$ - Quotient Rule

$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$ - Chain Rule

$\frac{d}{dx}\sin(x)=\cos(x)$

$\frac{d}{dx}\cos(x)=-\sin(x)$

$\frac{d}{dx}\tan(x)=\sec^2(x)$

$\frac{d}{dx}\cot(x)=-csc^2(x)$

$\frac{d}{dx}\arcsin(x)=\frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}\arccos(x)=-\frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}\arctan(x)=\frac{1}{1+x^2}$

$\frac{d}{dx}\text{arccot}(x)=-\frac{1}{1+x^2}$

$\frac{d}{dx}\text{arcsec}(x)=\frac{1}{x\sqrt{x^2-1}}$

$\frac{d}{dx}\text{arccsc}(x)=-\frac{1}{x\sqrt{x^2-1}}$

Other resources involving derivatives

Definition of a derivative Problems Calculus - forum

First Derivative Calculator(Solver) with Steps

Answer:

The Most Important Derivatives - Basic Formulas/Rules

Other resources involving derivatives