# First Derivative Calculator(Solver) with Steps

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

Function
$$\frac{d}{d x} e^{- 3 x \sin^{2}{\left (x \right )} + 1}$$

1. Let $$u = - 3 x \sin^{2}{\left (x \right )} + 1$$.

2. The derivative of $$e^{u}$$ is itself.

3. Then, apply the chain rule. Multiply by $$\frac{d}{d x}\left(- 3 x \sin^{2}{\left (x \right )} + 1\right)$$:

1. Differentiate $$- 3 x \sin^{2}{\left (x \right )} + 1$$ term by term:

1. The derivative of the constant $$1$$ 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:

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

$$f{\left (x \right )} = x$$; to find $$\frac{d}{d x} f{\left (x \right )}$$:

1. Apply the power rule: $$x$$ goes to $$1$$

$$g{\left (x \right )} = \sin^{2}{\left (x \right )}$$; to find $$\frac{d}{d x} g{\left (x \right )}$$:

1. Let $$u = \sin{\left (x \right )}$$.

2. Apply the power rule: $$u^{2}$$ goes to $$2 u$$

3. Then, apply the chain rule. Multiply by $$\frac{d}{d x} \sin{\left (x \right )}$$:

1. The derivative of sine is cosine:

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

The result of the chain rule is:

$$2 \sin{\left (x \right )} \cos{\left (x \right )}$$

The result is: $$2 x \sin{\left (x \right )} \cos{\left (x \right )} + \sin^{2}{\left (x \right )}$$

So, the result is: $$- 6 x \sin{\left (x \right )} \cos{\left (x \right )} - 3 \sin^{2}{\left (x \right )}$$

The result is: $$- 6 x \sin{\left (x \right )} \cos{\left (x \right )} - 3 \sin^{2}{\left (x \right )}$$

The result of the chain rule is:

$$\left(- 6 x \sin{\left (x \right )} \cos{\left (x \right )} - 3 \sin^{2}{\left (x \right )}\right) e^{- 3 x \sin^{2}{\left (x \right )} + 1}$$

4. Now simplify:

$$- \frac{3}{2} \left(2 x \sin{\left (2 x \right )} - \cos{\left (2 x \right )} + 1\right) e^{\frac{3 x}{2} \left(\cos{\left (2 x \right )} - 1\right) + 1}$$

$$- \frac{3}{2} \left(2 x \sin{\left (2 x \right )} - \cos{\left (2 x \right )} + 1\right) e^{\frac{3 x}{2} \left(\cos{\left (2 x \right )} - 1\right) + 1}$$

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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