Calculus Help! How to find the derivative?

Question

Calculate the derivative for
f(x) = x^3/2(2x^4 − 5x + x^−1/2)

Answer 1

f(x) = [x³/2][2x⁴ - 5x + x^-1/2]

f(x) = [x³/2][2x⁴ - 5x + 1/√x]

f(x) = [x³/2][2x⁴] - [x³/2][5x] + [x³/2][x^-1/2]

Exponent Laws: x^mxⁿ = x^m+n

f(x) = [x⁷] - (5/2)x⁴ + [x³/2][x^-1/2]

f(x) = [x⁷] - (5/2)x⁴ + (1/2)[x^5/2]

Apply 'derivative with respect to x' each side.

f'(x) = (d/dx){[x⁷] - (5/2)x⁴ + (1/2)[x^5/2]}

f'(x) = (d/dx)[x⁷] - (d/dx)(5/2)x⁴ + (d/dx)(1/2)x^5/2

Power Rule of Derivative (d/dx)xⁿ = nx^n-1

f'(x) = 7x⁶ - (5/2)(4x³) + (1/2)(5/2)[x^3/2]

f'(x) = 7x⁶ - (5)(2x³) + (5/4)[x^3/2]

f'(x) = 7x⁶ - 10x³ + (5/4)x^3/2

Therefore the derivative of f(x) is 7x⁶ - 10x³ + (5/4)x^3/2.