Answer:
Let's find the derivatives of the given functions step by step:
1. Dx(8x - 31 + x^319):
The derivative of 8x with respect to x is 8.
The derivative of -31 with respect to x is 0 (since it's a constant).
To find the derivative of x^319, we can use the power rule. The power rule states that if you have a term x^n, its derivative is n*x^(n-1).
So, the derivative of x^319 with respect to x is 319*x^(319-1) = 319*x^318.
Now, add all the derivatives together:
Dx(8x - 31 + x^319) = 8 + 0 + 319*x^318 = 8 + 319*x^318.
2. h(x) = (x^9 - 2)^3:
To find the derivative of this function, we can use the chain rule. The chain rule states that if you have a function g(f(x)), then its derivative is g'(f(x)) * f'(x).
Let g(u) = u^3, and f(x) = x^9 - 2. Then, h(x) = g(f(x)).
The derivative of g(u) = u^3 with respect to u is 3u^2.
The derivative of f(x) = x^9 - 2 with respect to x is 9x^8.
Now, apply the chain rule:
h'(x) = g'(f(x)) * f'(x) = 3(f(x))^2 * 9x^8 = 27x^8 * (x^9 - 2)^2.
3. y = 3x - 12:
The derivative of a constant (in this case, -12) with respect to x is 0.
The derivative of 3x with respect to x is 3.
So, dy/dx = 3.
4. y = x^55 - x^44 + x^3 + 3:
To find the derivative of this function, we can apply the power rule to each term separately.
The derivative of x^55 with respect to x is 55x^(55-1) = 55x^54.
The derivative of x^44 with respect to x is 44x^(44-1) = 44x^43.
The derivative of x^3 with respect to x is 3x^(3-1) = 3x^2.
The derivative of a constant (3) with respect to x is 0.
Now, add all the derivatives together:
dy/dx = 55x^54 - 44x^43 + 3x^2.
So, the derivatives of the given functions are as follows:
1. Dx(8x - 31 + x^319) = 8 + 319*x^318.
2. h'(x) = 27x^8 * (x^9 - 2)^2.
3. dy/dx = 3.
4. dy/dx = 55x^54 - 44x^43 + 3x^2.
Explanation: