1, definition of derivative: When the increments of the independent variable Δ δx = x-x0 and Δ x→ 0, the limit of the ratio of the increment of the function Δ y = f (x)-f' (x0) to the increment of the independent variable exists and is limited, so the function f is derivable at x0, which is called the derivative (or rate of change) of f at x0. Function y = f (x
Generally speaking, we get the rule of judging monotonicity of function by derivative: let y = f (x) be derivable in (a, b). If f' (x) >: 0 in (a, b), then f(x) is monotonically increasing in this interval (the tangent slope of this point increases, and the function curve becomes "steep" and rises). If in (a, b), f' (x) < 0, then f(x) decreases monotonically in this interval. So when f'(x)=0, y=f(x) has a maximum or minimum, the maximum is the maximum and the minimum is the minimum. Find the derivative of the function y = f (x) at x0:
① Find the increment δ y = f (x0+δ x)-f (x0) of the function.
② Find the average change rate.
③ Seek the limit and derivative.
Derived formula:
① C'=0(C is a constant function);
②(x^n)'= nx^(n- 1)(n∈q *)； Remember the derivative of1/x.
③(sinx)' = cosx；
(cosx)' =-sinx；
(tanx)'= 1/(cosx)^2=(secx)^2= 1+(tanx)^2
-(cotx)'= 1/(sinx)^2=(cscx)^2= 1+(cotx)^2
(secx)'=tanx secx
(cscx)'=-cotx cscx
(arcsinx)'= 1/( 1-x^2)^ 1/2
(arccosx)'=- 1/( 1-x^2)^ 1/2
(arctanx)'= 1/( 1+x^2)
(arccotx)'=- 1/( 1+x^2)
(arcsecx)'= 1/(|x|(x^2- 1)^ 1/2)
(arccscx)'=- 1/(|x|(x^2- 1)^ 1/2)
④ (sin(hx))'=hcoshx
(cos(hx))'=-hsinhx
(tan(hx))'= 1/(cos(hx))^2=(sec(hx))^2
(cot(hx))'=- 1/(sin(hx))^2=-(csc(hx))^2
(seconds (hx))'=-tan(hx) seconds (hx)
(csc(hx))'=-cot(hx)
(arsin(hx))'= 1/(x^2+ 1)^ 1/2
(arcos(hx))'= 1/(x^2- 1)^ 1/2
(artan(hx))'= 1/(x^2- 1)(| x | & lt； 1)
(arcot(hx))'= 1/(x^2- 1)(| x | > 1)
(arsec(hx))'= 1/(x( 1-x^2)^ 1/2)
(arcsc(hx))'= 1/(x( 1+x^2)^ 1/2)
⑤(e^x)' = e^x；
(a x)' = a xlna (ln is the natural logarithm)
(Inx)' = 1/x(ln is natural logarithm)
(log(ax))‘=(xlna)^(- 1),(a>； 0 and a is not equal to 1)
(x^ 1/2)'=[2(x^ 1/2)]^(- 1)
( 1/x)'=-x^(-2)
Application of derivative:
Monotonicity of 1. function
(1) Using the sign of derivative to judge the increase or decrease of function is an application of the geometric meaning of derivative in studying the law of curve change. It fully embodies the idea of combining numbers with shapes. Generally speaking, if f' (x) > 0 is in a certain interval, then the function y=f(x) monotonically increases in this interval. If f' (x) < 0, the function y=f(x) monotonically decreases in this interval. If f'(x)=0 is in a certain interval, then f(x) is a constant function. Note: in a certain interval, f' (x) > 0 is a sufficient condition for f(x) to be increasing function in this interval. That is to say, if f(x) is known as increasing function, you must write f'(x)≥0 when solving the problem.
(2) The step of finding the monotone interval of the function (1. Define the most basic solution 2. Monotonicity of compound function) ① Determine the domain of f(x); ② Deduction; (3) Use (or) to solve the range corresponding to x. When f' (x) > 0, f(x) is increasing function in the corresponding interval; When f' (x) < 0, f(x) is a decreasing function in the corresponding interval.
2. Extreme value of function
Determination of extreme value of (1) function ① If both sides have the same sign, it is not the extreme point of f(x); ② If the signs on the left and right sides are different nearby, it is the maximum or minimum.
3. Steps to find the extreme value of the function
(1) Determine the functional domain; ② Deduction; (3) Find all the points on the domain where there are no stationary points and derivatives, that is, find all the real roots of the sum of equations; (4) Check the symbols around the stagnation point. If Zuo Zheng is negative to the right, then f(x) gets the maximum at this root; If the left is negative and the right is positive, then f(x) takes the minimum value at this root.
4. The maximum value of the function
(1) If the maximum (or minimum) of f(x) in [a, b] is obtained at a point in (a, b), it is obvious that this maximum (or minimum) is also a maximum (or minimum), which is all the maximum of f(x) in (a, b). ② Compare f(x) with the extreme values of f(a) and f(b), where the largest is the maximum value and the smallest is the minimum value.
5. Life optimization
In life, we often encounter problems such as maximizing profits, saving materials and achieving the highest efficiency. These problems are called optimization problems, and optimization problems are also called maximum problems. Solving these problems has important practical significance. These problems can usually be transformed into function problems in mathematics, and then into the problem of finding the maximum (minimum) value of the function.