Questions: Find the derivative of the following function. f(x) = 3x^4 + 9/x^5 - 8x + 10 A. f'(x) = 12x^3 + 9/x^5 - 8 B. f'(x) = 12x^3 - 45/x^6 C. f'(x) = 3x^3 - 45/x^5 + 2 D. f'(x) = 12x^3 - 45/x^6 - 8

Solution Steps

The function is \( f(x) = 3x^{4} + \frac{9}{x^{5}} - 8x + 10 \). To find the derivative \( f'(x) \), differentiate each term individually.

The derivative of \( 3x^{4} \) is: \[ \frac{d}{dx}(3x^{4}) = 3 \cdot 4x^{3} = 12x^{3}. \]

Rewrite \( \frac{9}{x^{5}} \) as \( 9x^{-5} \). The derivative is: \[ \frac{d}{dx}(9x^{-5}) = 9 \cdot (-5)x^{-6} = -\frac{45}{x^{6}}. \]

The derivative of \( -8x \) is: \[ \frac{d}{dx}(-8x) = -8. \]

The derivative of a constant is zero: \[ \frac{d}{dx}(10) = 0. \]

Combine the derivatives of all terms to get: \[ f'(x) = 12x^{3} - \frac{45}{x^{6}} - 8. \]

Final Answer