### Basic Derivatives

The derivative notation of the $y=f(x)$ function is: $y'$, $f'(x)$, $\displaystyle \frac{dy}{dx}$, and $D_x ~y$. The following material describes the basic derivative formula that is very easy to remember. The derivatives of all forms of the function can be found. In contrast to integrals, not all forms of functions can be integrated. The following is the basic formula for the derivative of the function.

1. $\displaystyle y=ax^n$ so $\displaystyle y'=anx^{n-1}$ where $a$ and $n$ are numbers.

2. $\displaystyle y=(f \pm g)(x)$ so $\displaystyle y'=(f' \pm g')(x)$

3. $\displaystyle y=(f.g)(x)$ so $\displaystyle y'=(f'.g+f.g')(x)$

4. $\displaystyle y=\left(\frac{f}{g}\right)(x)$ so $\displaystyle y'=\left(\frac{f'g-fg'}{g^2}\right)(x)$

5. $\displaystyle y=(f^n)(x)$ so $\displaystyle y'=(n.(f^{n-1}).f')(x)$

2. $\displaystyle y=(f \pm g)(x)$ so $\displaystyle y'=(f' \pm g')(x)$

3. $\displaystyle y=(f.g)(x)$ so $\displaystyle y'=(f'.g+f.g')(x)$

4. $\displaystyle y=\left(\frac{f}{g}\right)(x)$ so $\displaystyle y'=\left(\frac{f'g-fg'}{g^2}\right)(x)$

5. $\displaystyle y=(f^n)(x)$ so $\displaystyle y'=(n.(f^{n-1}).f')(x)$

**Example:**

Given function $f(x)=3x^2+5$ and $g(x)=-x+1$. Define:

1. $\displaystyle D_x f^3$

2. $\displaystyle (f.g^2)'(2)$

**Solution:**

1. $\displaystyle =3.(3x^2+5)^2.(6x)=18x.(3x^2+5)^2$

2. $\displaystyle D_x g^2=-2(-x+1)=2x-2$ than

$\displaystyle D_x f.g^2=(6x)(-x+1)^2+(3x^2+5)(2x-2)$

$\displaystyle (f.g^2)'(2)=12.(-1)^2+(17)(2)=46$.

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