How to export 1 composite function
Rule: 1, let u=g(x), and deduce f(u) as follows: f' (x) = f' (u) * g' (x);
2. Let u = g (x) and a = p (u) and derive f (a): f' (x) = f' (a) * p' (u) * g' (x);
Expand:
1, let the domain of function y=f(u) be Du, that of function u=g(x) be Dx, and that of function u = g (x) be Mx. If Mx∩Du≦? , then any x in Mx∩Du passes through u; If there is a uniquely determined value of y corresponding to it, then a functional relationship between variables X and Y is formed by variable U, which is called a composite function, and it is recorded as: y=f[g(x)], where X is called an independent variable, U is an intermediate variable and Y is a dependent variable (i.e. a function).
2. domain: if the domain of function y=f(u) is b and that of u = g (x) is a, the domain of compound function y=f[g(x)] is D= {x|x∈A, and g(x)∈B} is considered comprehensively.
3. Periodicity: Let the minimum positive period of y=f(u) be T 1 and the minimum positive period of μ = φ (x) be T2, then the minimum positive period of y=f(μ) is T 1*T2, and any period can be expressed as k * t1* t2 (.
4. Determinants of monotonicity (increase or decrease): determined by monotonicity of y=f(u) and μ=φ(x). That is, "increase+increase = increase; Minus+Minus = Increase; Increase+decrease = decrease; Minus+increase = decrease "can be simplified as" same increase but different decrease ".
Derivation rule of 1 composite function
Y=f(u), U=g(x), then y'=f(u)'*g(x)'
Example 1.y = ln (x 3), y = ln (u), u = x 3,
y′=f(u)′*g(x)′=[ 1/ln(x^3)]*(x^3)′=[ 1/ln(x^3)]*(3x^2)
=(3x^2)/Ln(x^3)]
Example 2.y=cos(x/3), Y=cosu, u=x/3.
The derivation rule of y =-sin (x/3) * (1/3) =-sin (x/3)/3 by composite function.
What are the properties of 1 composite function?
The properties of a composite function are determined by the properties of the functions that make it up, and have the following laws:
(1) monotonicity law
If the function u=g(x) is monotonic in the interval [m, n] and the function y=f(u) is monotonic in the interval [g(m), g(n)] (or [g(n), g(m)], then
If u=g(x) and y=f(u) have the same increase or decrease, then the composite function y=f[g(x)] is increasing function; If u=g(x) and y= f(u) are different, then y=f[g(x)] is a decreasing function.
(2) parity law
If the definition domains of functions g(x), f(x) and f[g(x)] are all symmetrical about the origin, then u=g(x), y=f(u) are all odd function, and y=f[g(x)] is odd function; U=g(x), y=f(u) is an even function, or when it is odd and even, y= f[g(x)] is even.