Question

Найти D(у),если: а)у=4х в квадрате - 5х -1

Answer 1

Y=4√x-5x-1;x€[0;+oo)

1)y'=4*1/2*1/√x-5=0
(2/√x)-5=0
(2-5√x)/√x=0
√x≠0;x≠0
2-5√x=0
5√x=2
√x=2/5
x=4/25
2)y'>0
{√x>0;x€(0;+oo)
{2-5√x≥0;√x≤2/5;x€(0;4/25)]
3)y'<0
{√x>0;x€(0;+oo)
{2-5√x≤0
5√x≥2
√x≥2/5
x≥4/25
x€[4/25;+oo)
4)0____+___4/25____-__
x(max)=4/25
5)y(4/25)=4•2/5-5•4/25-1=
8/5-4/5-1=4/5-1=(4-5)/5=
=-1/5
6)y(0)=-1
7)E(y)=(-oo;-1/5)

Answer 2

To find the derivative D(y) of the function y = 4x^2 - 5x - 1 with respect to x, we'll apply the rules of differentiation. The derivative gives us the rate of change of the function with respect to x.

The general power rule for differentiation states that if we have a function of the form f(x) = x^n, the derivative is given by D(f(x)) = n * x^(n-1).

Now, let's find the derivative D(y) step by step:

1. Differentiate 4x^2 with respect to x: D(4x^2) = 2 * 4x^(2-1) = 8x

2. Differentiate -5x with respect to x: D(-5x) = -5

3. The derivative of a constant (in this case, -1) is zero: D(-1) = 0

Now, we add up the derivatives of each term to get the final result:

D(y) = D(4x^2) - D(5x) - D(1) = 8x - 5 - 0 = 8x - 5

So, the derivative D(y) with respect to x is 8x - 5.