Questions: Find the derivative of (0.4/(x^(1/4)) - 3.8x^(-4) + 2x) with respect to x.

Transcript text: Find $\frac{d}{d x}\left(\frac{0.4}{\sqrt[4]{x}}-3.8 x^{-4}+2 x\right)$.

Solution

Solution Steps

To find the derivative of the given function, we will apply the power rule for differentiation. The function is a combination of terms with different powers of $x$. We will differentiate each term separately and then combine the results.

Step 1: Differentiate Each Term

To find the derivative of the function $ f(x) = \frac{0.4}{\sqrt[4]{x}} - 3.8 x^{-4} + 2x $, we differentiate each term separately using the power rule.

The derivative of $ \frac{0.4}{\sqrt[4]{x}} $ can be rewritten as $ 0.4 x^{-1/4} $. Differentiating this term gives: \[ \frac{d}{dx}\left(0.4 x^{-1/4}\right) = 0.4 \times \left(-\frac{1}{4}\right) x^{-5/4} = -0.1 x^{-5/4} \]
The derivative of $ -3.8 x^{-4} $ is: \[ \frac{d}{dx}\left(-3.8 x^{-4}\right) = -3.8 \times (-4) x^{-5} = 15.2 x^{-5} \]
The derivative of $ 2x $ is: \[ \frac{d}{dx}(2x) = 2 \]

Step 2: Combine the Derivatives

Combine the derivatives of each term to find the derivative of the entire function: \[ f'(x) = -0.1 x^{-5/4} + 15.2 x^{-5} + 2 \]