Determine the first second and third derivatives of 2x^13-3x^8+x^7-5x^3
The given function is f(x) = 2x^13 - 3x^8 + x^7 - 5x^3.
To find the first derivative, we differentiate each term of the function:
f'(x) = d(2x^13)/dx - d(3x^8)/dx + d(x^7)/dx - d(5x^3)/dx
= 26x^12 - 24x^7 + 7x^6 - 15x^2
To find the second derivative, we differentiate each term of the first derivative:
f''(x) = d(26x^12)/dx - d(24x^7)/dx + d(7x^6)/dx - d(15x^2)/dx
= 312x^11 - 168x^6 + 42x^5 - 30x
To find the third derivative, we differentiate each term of the second derivative:
f'''(x) = d(312x^11)/dx - d(168x^6)/dx + d(42x^5)/dx - d(30x)/dx
= 3432x^10 - 1008x^5 + 210x^4 - 30
To determine the derivatives of the given function, we apply the power rule for differentiation. The power rule states that if we have a term of the form ax^n, then its derivative is given by d/dx(ax^n) = anx^(n-1).
Let's find the derivatives step-by-step:
First derivative:
The first derivative of the function 2x^13-3x^8+x^7-5x^3 is obtained by differentiating each term separately:
d/dx (2x^13 - 3x^8 + x^7 - 5x^3)
= 2*d/dx(x^13) - 3*d/dx(x^8) + d/dx(x^7) - 5*d/dx(x^3)
= 2*13x^(13-1) - 3*8x^(8-1) + 7x^(7-1) - 5*3x^(3-1)
= 26x^12 - 24x^7 + 7x^6 - 15x^2
Second derivative:
To find the second derivative, we take the derivative of the first derivative:
d/dx (26x^12 - 24x^7 + 7x^6 - 15x^2)
= 26*d/dx(x^12) - 24*d/dx(x^7) + 7*d/dx(x^6) - 15*d/dx(x^2)
= 26*12x^(12-1) - 24*7x^(7-1) + 7*6x^(6-1) - 15*2x^(2-1)
= 312x^11 - 168x^6 + 42x^5 - 30x
Third derivative:
To find the third derivative, we take the derivative of the second derivative:
d/dx (312x^11 - 168x^6 + 42x^5 - 30x)
= 312*d/dx(x^11) - 168*d/dx(x^6) + 42*d/dx(x^5) - 30*d/dx(x)
= 312*11x^(11-1) - 168*6x^(6-1) + 42*5x^(5-1) - 30
= 3432x^10 - 1008x^5 + 210x^4 - 30
Therefore, the first derivative of the given function is 26x^12 - 24x^7 + 7x^6 - 15x^2, the second derivative is 312x^11 - 168x^6 + 42x^5 - 30x, and the third derivative is 3432x^10 - 1008x^5 + 210x^4 - 30.