Questions: Find the derivative of (3.2/(x^(3/4)) - 4.1x^(-4) + 3x) with respect to x.

Transcript text: Find $\frac{d}{d x}\left(\frac{3.2}{\sqrt[4]{x^{3}}}-4.1 x^{-4}+3 x\right)$ \[ \frac{d}{d x}\left(\frac{3.2}{\sqrt[4]{x^{3}}}-4.1 x^{-4}+3 x\right)= \]

Solution

Solution Steps

Step 1: Rewrite the Function

We start with the function: \[ f(x) = \frac{3.2}{\sqrt[4]{x^{3}}} - 4.1 x^{-4} + 3 x \] This can be rewritten as: \[ f(x) = 3.2 x^{-3/4} - 4.1 x^{-4} + 3 x \]

Step 2: Differentiate Each Term

We apply the power rule to differentiate each term:

For $3.2 x^{-3/4}$, the derivative is: \[ \frac{d}{dx}(3.2 x^{-3/4}) = -\frac{2.4}{x^{7/4}} \]
For $-4.1 x^{-4}$, the derivative is: \[ \frac{d}{dx}(-4.1 x^{-4}) = 16.4 x^{-5} \]
For $3 x$, the derivative is: \[ \frac{d}{dx}(3 x) = 3 \]

Step 3: Combine the Derivatives

Now, we combine the derivatives of each term to find the overall derivative: \[ f'(x) = -\frac{2.4}{x^{7/4}} + 16.4 x^{-5} + 3 \] This can be expressed as: \[ f'(x) = -\frac{2.4}{x^{7/4}} + \frac{16.4}{x^{5}} + 3 \]