Derivative of 1. constant: for any constant c, there is f'(x)=0. This means that the slope of the constant function is 0, that is, it is constant at any point.
2. Derivative of power function: For power function f (x) = x n, there is f' (x) = n * x (n- 1). This means that the slope of the power function is proportional to the exponent.
3. Derivative of exponential function: For exponential function f (x) = e x, there is f' (x) = e x. This means that the slope of exponential function is equal to its own value.
4. Derivative of logarithmic function: for logarithmic function f(x)=log_a(x), there is f'(x)= 1/(x*ln(a)). This means that the slope of the logarithmic function is inversely proportional to the product of the base and the real number.
5. Derivative of trigonometric function: for sine function f(x)=sin(x), there is f' (x) = cos (x); For cosine function f(x)=cos(x), there is f' (x) =-sin (x); For the tangent function f(x)=tan(x), there is f' (x) = sec 2 (x). This means that the derivative of trigonometric function can be obtained by derivation.
6. Derivative of compound function: For compound function f(g(x)), there is f'(g(x))* g'(x). This means that the derivative of the composite function is equal to the derivative of the external function multiplied by the derivative of the internal function.
7. Derivative of implicit function: For implicit function F(x, y)=0, there is f' _ x (x, y) =-f' _ y (x, y)/f' _ xy (x, y). This means that the derivative of implicit function can be obtained by partial derivative.
8. Higher-order derivative: For any order n, there is f n (x) = (f' (x)) n+n * f (n-1) (x) * f' (x)+... This means that the higher-order derivative can be obtained through multiple derivatives.
9. Chain rule: For the compound function f(g(h(x)), there is f' (g (h (x)) = f' (g) * g' (h (x)) * h' (x). This means that the chain rule can be used to solve the derivative of complex composite functions.
10. Differential rule: for the case of adding or subtracting two functions, there is f' (a+b) = f' (a)+f' (b)-f' (a) * f' (b); F'(a-b)=f'(a)-f'(b). This means that when multiple functions are added or subtracted, the differential law can be used to solve the derivative.