Question

ДАЮ 40 БАЛЛОВ! Вычислить производную функции f(x)=2+ в точках 2; 4; х; х-3.

Answer 1

Ответ:

$20; \quad 64; \quad 4x+3x^{2}; \quad 3x^{2}-14x+15;$

Объяснение:

$f(x)=2x^{2}+x^{3};$

$f'(x)=(2x^{2}+x^{3})'=(2x^{2})'+(x^{3})'=2 \cdot (x^{2})'+3 \cdot x^{3-1}=2 \cdot 2 \cdot x^{2-1}+3x^{2}=$

$=4x+3x^{2};$

$f'(2)=4 \cdot 2+3 \cdot 2^{2}=8+3 \cdot 4=20;$

$f'(4)=4 \cdot 4+3 \cdot 4^{2}=16+3 \cdot 16=64;$

$f'(x-3)=4(x-3)+3(x-3)^{2}=4x-12+3(x^{2}-6x+9)=4x-12+3x^{2}-18x+27=$

$=3x^{2}-14x+15;$

Answer 2

Calculating the Derivative of the Function

To calculate the derivative of the function \( f(x) = 2x \) at the given points \( x = 2 \) and \( x = 4 \), we can use the power rule of differentiation, which states that the derivative of \( x^n \) is \( nx^{n-1} \).

1. Derivative at x = 2: - Using the power rule, the derivative of \( f(x) = 2x \) is \( f'(x) = 2 \). - Therefore, the derivative of the function at \( x = 2 \) is \( f'(2) = 2 \).

2. Derivative at x = 4: - Using the power rule, the derivative of \( f(x) = 2x \) is \( f'(x) = 2 \). - Therefore, the derivative of the function at \( x = 4 \) is \( f'(4) = 2 \).

Additional Points

For the additional points \( x \) and \( x-3 \), the derivative of the function \( f(x) = 2x \) remains constant at 2, as the coefficient of \( x \) is 2, and the power rule states that the derivative of a constant times \( x \) is the constant itself.

Therefore, for \( x \) and \( x-3 \), the derivative of the function \( f(x) = 2x \) is also 2.