What is the fourth derivative of this fifth degree polynomial? 5x^5+2x^4+7x^3+x^2+x+42?
first deriv. : 25x^4 + 8x^3 + 21x^2 + 2x + 1
2nd deriv. : 100x^3 + 24x^2 + 42x + 2
3rd ...
4th ...
If you know the pattern for taking derivatives of simple terms like those above, you should have no difficulty finishing the question.
To find the fourth derivative of a polynomial, we need to apply the power rule for differentiation multiple times. The power rule states that the derivative of x^n is n*x^(n-1).
Let's differentiate the polynomial step by step:
The given polynomial is: 5x^5 + 2x^4 + 7x^3 + x^2 + x + 42.
1. First derivative:
Take the derivative of each term:
d/dx (5x^5) = 5*5x^(5-1) = 25x^4.
d/dx (2x^4) = 4*2x^(4-1) = 8x^3.
d/dx (7x^3) = 3*7x^(3-1) = 21x^2.
d/dx (x^2) = 2x^(2-1) = 2x.
d/dx (x) = 1.
Combine all the derivatives to get the first derivative:
Derivative: 25x^4 + 8x^3 + 21x^2 + 2x + 1.
2. Second derivative:
Take the derivative of the first derivative:
d/dx (25x^4) = 4*25x^(4-1) = 100x^3.
d/dx (8x^3) = 3*8x^(3-1) = 24x^2.
d/dx (21x^2) = 2*21x^(2-1) = 42x.
d/dx (2x) = 2.
Combine all the derivatives to get the second derivative:
Derivative: 100x^3 + 24x^2 + 42x + 2.
3. Third derivative:
Take the derivative of the second derivative:
d/dx (100x^3) = 3*100x^(3-1) = 300x^2.
d/dx (24x^2) = 2*24x^(2-1) = 48x.
d/dx (42x) = 42.
Combine all the derivatives to get the third derivative:
Derivative: 300x^2 + 48x + 42.
4. Fourth derivative:
Take the derivative of the third derivative:
d/dx (300x^2) = 2*300x^(2-1) = 600x.
d/dx (48x) = 48.
Combine all the derivatives to get the fourth derivative:
Derivative: 600x + 48.
Therefore, the fourth derivative of the given fifth degree polynomial 5x^5 + 2x^4 + 7x^3 + x^2 + x + 42 is 600x + 48.