Derivative method of compound function

Derivative method of compound function

F [g (x)], let g (x) = u, then f [g (x)] = f (U), so (formula): F '[g (x)] = f' (U) * g '(x) give you an example
F [g (x)] = sin (2x), then let g (x) = 2x, let g (x) = 2x = u, then f (U) = sin (U), so f '[g (x)] = [sin (U)]' * (2x) '= 2cos (U), and then replace u with 2x to obtain f' [g (x)] = 2cos (2x)

Derivation of some composite functions 1. Y = 3 ^ 3-4x derivation 2. Y = sin {ln (4-x)} 3. Arccos root 2-3x 4. Y = lnsin root x ^ 3 + 1 5. Y = ln (x ^ 3 + 3 ^ x)

Derivation:
y=3^(3-4x)
y'=[3^(3-4x)](ln3)(-4)=-4ln3[3^(3-4x)]
y=sin[ln(4-x)]
y'={cos[ln(4-x)]}[-1/(4-x)]=[1/(x-4)]cos[ln(4-x)]
y=arccos√(2-3x)
y'=-{-3/[2√(2-3x)]}/√[1-(2-3x)]=3/{2√[(2-3x)(-1+3x)]}=3/[2√(-9x ²+ 9x-2)]
y=lnsin√(x ³+ 1)

About the derivation of composite functions ... the law of derivation Generally, the derivative y'x of the compound function y = f [g (x)] to the independent variable x is equal to the derivative y'u of the known function to the intermediate variable U = g (x) multiplied by the derivative u'x of the intermediate variable U to the independent variable x I look dizzy ... I'll catch up

I'll pass on my experience to you~
You just remember to take the derivative of a compound function
We must first distinguish who is and compound, that is, f (x) and G (x) in y = f [g (x)]
Let u = g (x) here
Then the derivative of Y is equal to the derivative of F (x) multiplied by the derivative of U

Derivation of compound function 1,y=sin4x 2, y = cosx2 (just the square of x) Please tell me why you did it

1. Let t = 4x, then y = Sint. Y '= dy / DX = (dy / DT) * (DT / DX) = (cost) * 4 = 4cos4x
2. Let t = x ^ 2, then y = cosx ^ 2. Y '= dy / DX = (dy / DT) * (DT / DX) = - Sint * 2x = - 2xsinx ^ 2

How to derive a compound function

Take the inner function as a whole, first take the derivative of the outer layer, then the derivative of the inner layer, and multiply the result

Derivative of compound function 1. Y = e to the power of 1 / x times SiNx 2. Y = sin (LNX) + sinxlnx

Derivation method (1) steps to find the derivative of function y = f (x) at x0: ① find the increment of the function Δ y=f(x0+ Δ x) - f (x0) ② calculate the average rate of change ③ take the limit to obtain the derivative. (2) derivative formulas of several common functions: ① C '= 0 (C is a constant); ② (x^n)'=nx^(n-1) (n∈Q)； ③ (sinx)'=c...