# Definition and Properties of the Derivative

• Functions: $$f$$, $$g$$, $$y$$, $$u$$, $$v$$
Argument (independent variable): $$x$$
Real constant: $$k$$
Angle: $$\alpha$$
1. The derivative of a function $$y = f\left( x \right)$$ measures the rate of change of $$y$$ with respect to $$x$$. Suppose that at some point $$x \in \mathbb{R}$$, the argument of a continuous real function $$y = f\left( x \right)$$ has an increment $$\Delta x$$. Then the increment of the function is equal to
$$\Delta y = f\left( {x + \Delta x} \right) \,-$$ $$f\left( x \right).$$
The derivative of the function $$y = f\left( x \right)$$ at the point $$x$$ is defined as the limit of the ratio $$\Delta y/\Delta x$$ as $$\Delta x \to 0:$$
$$y’ = f’\left( x \right) = {\large\frac{{dy}}{{dx}}\normalsize} =$$ $${\large\frac{{df\left( x \right)}}{{dx}}\normalsize} =$$ $$\lim\limits_{x \to 0} {\large\frac{{\Delta y}}{{\Delta x}}\normalsize} =$$ $$\lim\limits_{x \to 0} {\large\frac{{f\left( {x + \Delta x} \right) – f\left( x \right)}}{{\Delta x}}\normalsize}.$$
2. From a geometrical point of view, the derivative of a function $$y = f\left( x \right)$$ at a point $$x$$ is equal to the slope of the tangent line to the curve $$f\left( x \right)$$ drawn through this point:
$${\large\frac{{dy}}{{dx}}\normalsize} = \tan \alpha$$
3. The derivative of the sum of two functions is equal to the sum of their derivatives:
$${\large\frac{{d\left( {u + v} \right)}}{{dx}}\normalsize} = {\large\frac{{du}}{{dx}}\normalsize} + {\large\frac{{dv}}{{dx}}\normalsize}$$
4. The derivative of the difference of two functions is equal to the difference of their derivatives:
$${\large\frac{{d\left( {u – v} \right)}}{{dx}}\normalsize} = {\large\frac{{du}}{{dx}}\normalsize} – {\large\frac{{dv}}{{dx}}\normalsize}$$
5. A constant factor can be taken out of a derivative:
$${\large\frac{{d\left( {ku} \right)}}{{dx}}\normalsize} = k{\large\frac{{du}}{{dx}}\normalsize}$$
6. Derivative of the product of two functions
$${\large\frac{{d\left( {u \cdot v} \right)}}{{dx}}\normalsize} =$$ $${\large\frac{{du}}{{dx}}\normalsize} \cdot v + u \cdot {\large\frac{{dv}}{{dx}}\normalsize}$$
7. Derivative of the quotient of two functions
$${\large\frac{d}{{dx}}\normalsize}\left( {{\large\frac{u}{v}}\normalsize} \right) =$$ $${\large\frac{{\frac{{du}}{{dx}} \cdot v – u \cdot \frac{{dv}}{{dx}}}}{{{v^2}}}\normalsize}$$
8. Derivative of a composite function (chain rule)
$$y = f\left( {g\left( x \right)} \right),$$ $$u = g\left( x \right),$$ $${\large\frac{{dy}}{{dx}}\normalsize} = {\large\frac{{dy}}{{du}}\normalsize} \cdot {\large\frac{{du}}{{dx}}\normalsize}$$
9. Derivative of an inverse function
$${\large\frac{{dy}}{{dx}}\normalsize} = {\large\frac{1}{{dx/dy}}\normalsize},$$
where $$x\left( y \right)$$ is the inverse function for $$y\left( x \right).$$
10. $${\large\frac{d}{{dx}}\normalsize} \left( {{\large\frac{1}{y}}\normalsize} \right) =$$ $$– {\large\frac{{dy/dx}}{{{y^2}}}\normalsize}$$
11. Logarithmic derivative
$$y = f\left( x \right),$$ $$\ln y = \ln f\left( x \right),$$ $${\large\frac{{dy}}{{dx}}\normalsize} =$$ $$f\left( x \right) \cdot {\large\frac{d}{{dx}}\normalsize} \left[ {\ln f\left( x \right)} \right]$$