Question

G(x) = x^6 - 5x^3 +2x^-2 -7 h(x)=(x-1)*(x+2) Решите производную

Answer 1

G'(x)=6x^5-15x^2-4x^(-3)
h'(x)=(x-1)' *(x+2)+(x-1)*(x+2)'=1*(x+2)+1*(x-1)=2x+1

Answer 2

To find the derivative of the given functions G(x) and h(x), we'll take the derivative of each term separately using the basic rules of differentiation. Let's start with G(x):

Given function: G(x) = x^6 - 5x^3 + 2x^(-2) - 7

Step 1: Take the derivative of each term. d/dx (x^6) = 6x^(6-1) = 6x^5 d/dx (-5x^3) = -5 * 3x^(3-1) = -15x^2 d/dx (2x^(-2)) = 2 * -2x^(-2-1) = -4x^(-3) = -4/x^3 (Note: x^(-n) = 1/x^n) d/dx (-7) = 0 (the derivative of a constant is zero)

Step 2: Combine the derivatives of each term. G'(x) = 6x^5 - 15x^2 - 4/x^3

Next, let's find the derivative of h(x):

Given function: h(x) = (x - 1) * (x + 2)

To find h'(x), we can use the product rule, which states that if u(x) and v(x) are two functions, then the derivative of their product is given by:

(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

Step 1: Identify u(x) and v(x). u(x) = (x - 1) v(x) = (x + 2)

Step 2: Find u'(x) and v'(x). u'(x) = d/dx (x - 1) = 1 v'(x) = d/dx (x + 2) = 1

Step 3: Apply the product rule to find h'(x). h'(x) = u'(x) * v(x) + u(x) * v'(x) h'(x) = 1 * (x + 2) + (x - 1) * 1 h'(x) = x + 2 + x - 1 h'(x) = 2x + 1

So, the derivative of h(x) is h'(x) = 2x + 1.