Derivative

In calculus, derivatives measure the slope of a curve at any given point. This is the equivalent of measuring a curve's velocity at a given point.

In a given calculus equation, the derivative of a number n is notated as follows:
 * $$\frac{d}{dx} n = n'$$

The integral is the inverse of the derivative, as shown below:
 * $$\int n' dx = n + C$$

This is completely generalized in the Fundamental Theorem of Calculus:
 * $$\frac{d}{dx} \int^{x}_{a}f(t) dx = f(x)$$


 * $$\int^{b}_{a}F'(x) dx = F(b) - F(a)$$

Basic Properties

 * $$\frac{d}{dx} x^{n} = nx^{n-1}$$ for all x.


 * $$\frac{d}{dx} c = 0$$ for any constant number c.


 * $$\frac{d}{dx} cx = c$$ for any constant number c.


 * $$\frac{d}{dx} e^{x} = e^{x}$$ for all x, where $$e = \lim_{n \rightarrow \infty} (1 + \frac{1}{n}\,)^{n}$$.


 * $$\frac{d}{dx} a^{x} = a^{x} \ln{a}$$ for all ax ≠ 00.


 * $$\frac{d}{dx} \log_{a}{x} = \frac{1}{x\ln{a}}$$


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


 * $$\frac{d}{dx} x^{x} = x^{x} (\ln{x}+1)$$


 * $$\frac{d}{dx} uv = u'v + v'u$$. This is the product rule of derivatives.


 * $$\frac{d}{dx} (\frac{u}{v}) = \frac{u'v - v'u}{v^{2}} \iff v \neq 0$$. This is the quotient rule of derivatives.

Derivatives of Trig Functions

 * $$\frac{d}{dx} \sin{x} = \cos{x}$$


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


 * $$\frac{d}{dx} \tan{x} = \frac{1}{\cos^{2}{x}}$$


 * $$\frac{d}{dx} \cot{x} = -\frac{1}{sin^{2}{x}}$$


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


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


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


 * $$\frac{d}{dx} \arccot{x} = -\frac{1}{1+x^{2}}$$

Product and Quotient Rules of Derivatives
As seen above, the product rule of derivatives is
 * $$\frac{d}{dx} uv = u'v + v'u$$.

Here's an example of this rule in action.
 * $$\frac{d}{dx} (x^{2} * 2x^{3}) = (2x * 2x^{3}) + (6x^{2} * x^{2})$$


 * $$= 4x^{4} + 6x^{4} = 10x^{4}$$

Let's check this with an integral, since
 * $$\int n' dx = n + C$$.


 * $$\int 10x^{4} dx = 10\int x^{4} dx$$
 * $$= \frac{10x^{5}}{5} + C$$
 * $$= 2x^{5} + C = x^{2} * 2x^{3} + C$$

As seen above, the quotient rule of derivatives is
 * $$\frac{d}{dx} (\frac{u}{v}) = \frac{u'v - v'u}{v^{2}} \iff v \neq 0$$.

Here's an example of this rule in action.
 * $$\frac{d}{dx} (\frac{3x^{2}}{4x^{3}}) = \frac{(6x * 4x^{3}) - (12x^{2} * 3x^{2})}{16x^{6}}$$


 * $$= \frac{24x^{4}}{16x^{6}} - \frac{36x^{4}}{16x^{6}} = -\frac{12x^{4}}{16x^{6}} = -\frac{3}{4x^{2}}$$

Again, let's verify this using an integral.
 * $$\int-\frac{3}{4x^{2}} dx = -\frac{3}{4}\int x^{-2} dx$$


 * $$= \frac{3}{4x} + C = \frac{3x^{2}}{4x^{3}} + C$$