How to find the derivative of a composite function, how to find the derivative of a composite functi

1.Let u=g(x), and for f(u) we derive: f'(x)=f'(u)*g'(x);2.

Let u=g(x) and a=p(u), and for f(a) we derive: f'(x)=f'(a)*p'(u)*g'(x);

Let the function y=f(u) define the domain.

is du, the value range.

is mu, and the domain of the function u=g(x) is d.

m du≠ , then for any x in m du, pass u; If there is a uniquely determined value of y, then the variable x and y form a functional relationship through the variable u, which is called a composite function.

compositefunction)。

Generally, it is carried out in the following three steps:

1) Appropriately select intermediate variables and correctly decompose composite relationships;

2) step-by-step derivation (figuring out which variable is derivative of which variable is derivative of which variable at each step);

3) Replace the intermediate variable back with the original independent variable.

Generally x).

That is to say, firstly, the intermediate variables are selected, the composite relationship is decomposed, and the functional relationship y=f( )=f(x) is explained. Then the known function is derived from the intermediate variable, and the intermediate variable is derived from the independent variable. Finally, find and substitute the intermediate variable back as a function of the independent variable. The whole process can be simply recorded as decomposition-derivative-regeneration. Once you are proficient, you can omit the intermediate process.

In the case of multiple compounds, intermediate variables can be used multiple times accordingly.

f(x)=(1-x) 5+(1+x) 5.

1-x) is -1 and (1+x) is 1f'(x)=-1*5(1-x)^4+1*5(1+x)^45(1+x)^4-5(1-x)^4

Derivative formula for composite functions.

How to find the rules for the derivative of composite functions:

1. Let u=g(x), and find the derivative of f(u) to obtain: f'(x)=f'(u)*g'(x)；

2. Let u=g(x) and a=p(u), and find the derivative of f(a) as f'(x)=f'(a)*p'(u)*g'(x)。

Definition. Let the domain of the function y=f(u) be du and the range of values be mu, and the domain of function u=g(x) be dx and the range of mx, if mx du ≠, then for any x in mx du pass u; If there is a uniquely determined value of y, then the variable x and y form a functional relationship through the variable u, which is called the composite function, denoted as: y=f, where x is called the independent variable, u is the intermediate variable, and y is the dependent variable (i.e., the function).

Domain. If the domain of the function y=f(u) is b and the domain of u=g(x) is a, then the domain of the composite function y=f is d= consider the range of x values of each part and take their intersection.

Periodicity. Let the minimum positive period of y=f(u) be t1, and the minimum positive period of x) is t2, then the minimum positive period of y=f( ) is t1*t2, and any period can be expressed as k*t1*t2 (k belongs to r+).4. Determinants of monotony (increase or decrease):

It is determined by the monotonicity of y=f(u), x).

i.e. "increase increase; Decrease, reduce, touch, tease, increase; increase or decrease; "Decrease, increase, decrease", which can be simplified to "increase with the same and decrease from the difference".

The derivative of the composite function is 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) x 3) 3x 2)=(3x 2) ln(x 3)].

2. y=cos(x 3),y=cosu,u=x 3 y=-sin(x 3)*(1 3)=sin(x 3) 3 is obtained by the composite function to find the derivative of the joke.

What are the properties of a composite functionThe properties of a composite function are determined by the properties of the functions that compose it, and have the following laws:

1) Monotonicity If the function u=g(x) is monotonic in the interval m,n, and the function y=f(u) is also 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 incrementality, then the composite function y=f is an increasing function; If u=g(x) and y=f(u) have different increases and decreases, then y=f is a subtraction function.

2) Parity law: If the domains of the functions g(x), f(x), and f are all symmetrical about the origin, then u=g(x), y=f(u) are all odd functions, and y=f is an odd function; u=g(x) and y=f(u) are even functions, or in the case of odd and even, y= f is an even function.

The derivative of composite functions can be calculated using the chain rule of derivatives.

Solution: 1. Analyze y=sin[ln(2x+3)] as being caused by multiple functions, i.e.

y=sin(u), u=ln(v), v=2x+32, derivatives respectively.

dy/du=cos(u)

du/dv=1/v

dv/dx=2

3. Use the chain rule to calculate DY DX

dy dx=dy du·du dv·dv dx4, finally, put u,v back to the above formula, and get the result.

Solution: <>

Extended knowledge: The chain rule is the derivative of the tremor in calculus, which is used to find the derivative (partial derivative) of a composite function, and is a common method in the derivative operation of calculus. The derivative of the composite function will be the product of the derivatives of the finite functions that constitute the composite at the corresponding point, just like a chain, one ring imitates the first set of rings, so it is called the chain rule.

The chain rule for the derivative of unary functions.

The chain rule for derivatives of multivariate functions.

The derivative of the composite function is as follows: if the function u=g(x) is derivable at the point x, and y=f(u) is derivable at the point u=g(x), then the composite function y=f[g(x)] is derivable at the point x, and its derivative is dy dx=f'(u)·g'(x) or dy dx = (dy du) · (du dx)。

Let the domain of the function y=f(u) be du and the value range be mu, and the domain of the number u=g(x) is dx and the value range be mx, if mx du ≠ , then for any x in mx du pass u; If there is a uniquely determined value of y, then there is a functional relationship between the variable x and y through the variable u, which is called the composite function, denoted as: y=f[g(x)], where x is called the independent variable, u is the intermediate variable, and y is the dependent variable (i.e., the function).

Derivative is a mathematical method of calculation that is defined as the limit of the quotient between the increment of the dependent variable and the increment of the independent variable when the increment of the independent variable tends to zero. When there is a derivative of a function, it is said to be derivable or differentiable. The derivable function must be continuous.

Discontinuous functions must not be derivative.

Derivation is the foundation of calculus and an important pillar of calculus calculations. Some important concepts in physics, geometry, economics, and other disciplines can be represented by derivatives. For example, the derivative can represent the instantaneous velocity and acceleration of a moving object, the slope of a curve at a point, and the marginality and elasticity in economics.

<> first find the derivative of the function y=ln(2x+1):

y‘=(2x+1)*[1/(2x+1)]。

2/(2x+1)

Molecule (sinx) (n+1).

Denominator (n+1) cosx

Yu Nianhuai accompanies the strings:

Molecule (cosx) (n+1).

Denominator - (n+1)sinx Supplementary answer The definite integral is the original function of the derivative function, (sinx) n is a composite function, you can first calculate the original function of t n, and then compound sinx=t. The thought process is stupid:

t) The derivative of (n+1) is (n+1)*t n, so the original function is to be divided by one (n+1).

Then t=sinx, the derivative of sinx is cosx, so the original function should be divided by one cosx

I don't say much clearly.

If h(x)=f(g(x)), then h'(x)=f'(g(x))g'(x).

The chain method and the world are the derivative rules in calculus, which are used to find the derivative of a composite function, and are a common method in derivative operations of calculus. The derivative of the composite function will be the product of the derivatives of the finite functions that make up the composite at the corresponding point, just like a chain, so it is called the chain rule.

The chain rule is the law for finding the derivatives (partial derivatives) of a composite function, if i and j are the open intervals on a straight line, the function f(x) is differentiated where i is defined, the function g(y) is defined on j and differentiable at f(a), then the composite function is differentiable at a (defined on i), and moduloIf u=g(y) and y=f(x), and f is differentiable on i and g is differentiable on j, then on i any call limb x has.

How to find the bridge derivative of the missing number of the first part of the composite function?

To calculate the derivative of a composite function, it is necessary to use the chain rule: the chain law states that the derivative of the composite function is the product of the respective derivatives when the function is taken as a parameter between a continuous function and other functions. Thus, the derivative of a composite function can be expressed as the derivative of the product of the original function and other functions.

withTaylor's formulaPlacing cosx at x0=0 yields:

cosx=1-x^2/2+x^4/4-x^6/6+..1)^nx^2n/2n...

Thus 1-cosx=x 2 2-x 4 4+x 6 6+...1)^nx^2n/2n...

Therefore x2 2 is the main part of 1-cosx.

Therefore, lim[(1-cosx) (x 2 2)]=1(x 0), and from the definition of equivalent infinitesimal quantities, we can see that 1-cosx and x 2 2 are equivalent infinitesimal quantities, i.e., cosx-1 and -(x 2) 2 are equivalent infinitesimal quantities. Spinal order.

1. Composite functions.

Derivative.

Composite function vs. independent variable.

, which is equal to the derivative of the known function to the intermediate variable, multiplied by the derivative of the intermediate variable to the independent variable.

i.e. for y=f(t), t=g(x), then y'The formula is expressed as: y'=（f(t)）'Sakura Jun*(g(x)).'

For example: y=sin(cosx), then y'=cos(cosx)*(sinx)=-sinx*cos(cosx)

2、（lnx）'=1/x、（e^x）'=e^x、（c）'=0 (c is constant).

3. The four rules of operation of derivatives are dissolved.

1）（f(x)±g(x)）'f'(x)±g'(x)

Example: (x 3-cosx).'=x^3)'-cosx)'=3*x^2+sinx

2）（f(x)*g(x)）'f'(x)*g(x)+f(x)*g'(x)

Example: (x*cosx).'=x)'*cosx+x*(cosx)'=cosx-x*sinx