Question

Find the derivative of \( y = x^6 + 48x^3 + 24x - 16 \) with respect to \( x \).

Hint:

When differentiating terms with powers of \( x \), apply the power rule: \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \).

Answer 1

To find the derivative of the given function, we will apply the power rule for differentiation, which states that for \( y = ax^n \), the derivative is \( \frac{dy}{dx} = n \cdot ax^{n-1} \). Given: \[ y = x^6 + 48x^3 + 24x - 16 \] We will differentiate each term of the function with respect to \( x \). \[ \frac{dy}{dx} = \frac{d}{dx}(x^6) + \frac{d}{dx}(48x^3) + \frac{d}{dx}(24x) - \frac{d}{dx}(16) \] Step 1: Differentiating each term.
1. The derivative of \( x^6 \) is: \[ \frac{d}{dx}(x^6) = 6x^5 \] 2. The derivative of \( 48x^3 \) is: \[ \frac{d}{dx}(48x^3) = 3 \times 48x^2 = 144x^2 \] 3. The derivative of \( 24x \) is: \[ \frac{d}{dx}(24x) = 24 \] 4. The derivative of \( -16 \) is: \[ \frac{d}{dx}(-16) = 0 \]
Step 2: Combine the derivatives.
Now, combine all the derivatives: \[ \frac{dy}{dx} = 6x^5 + 144x^2 + 24 \] Thus, the derivative of \( y \) with respect to \( x \) is: \[ \boxed{\frac{dy}{dx} = 6x^5 + 144x^2 + 24} \]