Questions: Calculate each derivative. If f(x) = 1/x^4 + 1/x^7 then f'(x) = -(4 / x^5) - (7 / x^8) If g(x) = -1/(3 x^0.3) then g'(x) = 0.1 / x^1.3 If h(x) = 3/x - 3/x^7 + 1/x^3 then h'(x) = -(3 / x^2) + (21 / x^8) - (3 / x^4)

Solution Steps

To calculate the derivatives of the given functions, we will use the power rule for differentiation. The power rule states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \). We will apply this rule to each term in the functions provided.

Given the function \[ f(x) = \frac{1}{x^4} + \frac{1}{x^7} = x^{-4} + x^{-7} \] we apply the power rule to find its derivative: \[ f'(x) = -4x^{-5} - 7x^{-8} = -\frac{4}{x^5} - \frac{7}{x^8} \]

For the function \[ g(x) = -\frac{1}{3x^{0.3}} = -\frac{1}{3} x^{-0.3} \] we again use the power rule: \[ g'(x) = -\frac{1}{3} \cdot (-0.3) x^{-1.3} = \frac{0.1}{x^{1.3}} \]

The function \[ h(x) = \frac{3}{x} - \frac{3}{x^7} + \frac{1}{x^3} = 3x^{-1} - 3x^{-7} + x^{-3} \] is differentiated as follows: \[ h'(x) = -3x^{-2} + 21x^{-8} - 3x^{-4} = -\frac{3}{x^2} + \frac{21}{x^8} - \frac{3}{x^4} \]

Final Answer