First Derivative Calculator(Solver) with Steps

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

Function


  1. Let [tex]u = - 3 x \sin^{2}{\left (x \right )} + 1[/tex].

  2. The derivative of [tex]e^{u}[/tex] is itself.

  3. 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]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]:

            1. The derivative of sine is cosine:

            The result of the chain rule is:

          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:

  4. Now simplify:


The answer is:

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


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