Power rule - Solved Examples

Question: `f(x) = x^4 - 3x^3+ 2x^2 + x - 1`
Solution:

`f'(x) = d/(dx) x^4 - 3x^3 + 2x^2 + x - 1`

`f'(x) = d/(dx) x^4 - d/(dx)3x^3 + d/(dx)2x^2 + d/(dx)x - d/(dx)1`

 Applying power rule on each term above,

`f'(x) = 4x^3 - 3(3x^2) + 2(2x) + 1 - 0`

Note that derivative of x is 1 because `d/(dx) x = d/(dx) x^1 = 1x^0 = 1`, and derivative of any constant is 0 because `d/(dx) k = d/(dx) kx^0 = d/(dx) k(0x^(-1)) = 0`
Simplify the above expression,

`f'(x) = 4x^3 - 9x^2 + 4x + 1`

Question: `f(x) = 5x^-5 + 2x^-3 + x^-1 + 5`
Solution:

`f'(x) = d/(dx) ( x^-5 + x^-3 + x^-1 + 5 )`
`f'(x) = d/(dx)  5x^-5 - d/(dx)  2x^-3 + d/(dx) x^-1 + d/(dx) 5`

Factor out constants,

`f'(x) = 5 d/(dx) x^-5 - 2 d/(dx) x^-3 + d/(dx) x^-1 + d/(dx) 5`

Apply power rule on each term,

`f'(x) = 5 (-5x^-6) - 2 (-3x^-4) + (-1x^-2) + 0`

Simplify the expression,

`f'(x) = -25x^-6 + 6x^-4 - x^-2`

Question: `f(x) = sqrt(x) + sqrt(x^3) + 1/x^(1/10)`
Solution:
Rewrite radicals as rational exponents,

`f(x) = x^(1/2) + x^(3/2) + x^(-1/10)`

Apply power rule on each term,

`f'(x) = (1/2)x^(1/2 - 1) + (3/2)x^(3/2 - 1) + (-1/10)x^(-1/10 - 1)`

Simplify the expression,

`f'(x) = (1/2)x^(-1/2) + (3/2)x^(1/2) - (1/10)x^(-11/10)`