Question - Determining the Slope of a Tangent to a Curve Involving Radicals

First, let's recall that the slope of the tangent line to a curve at a certain point is the value of the derivative of the function at that point. The derivative \( f'(x) \) gives us a formula to find the slope at any \( x \).

The derivative of the first term \( x^4 \) is straightforward; applying the power rule, which says to bring down the exponent and subtract one from it, we get \( 4x^3 \).

Now for the second term \(\frac{4}{\sqrt[3]{x^2}}\), we first rewrite it in a more derivative-friendly way, using exponents instead of roots. Recall that \( \sqrt[3]{x^2} = x^{2/3} \), so the term can be rewritten as \( 4x^{-2/3} \).

Using the power rule again, for \( 4x^{-2/3} \), we bring down the exponent, multiply it by the coefficient (which is 4), and then subtract one from the exponent. This gives us \( -4 \cdot \frac{2}{3}x^{-2/3 - 1} \) or \( -4 \cdot \frac{2}{3}x^{-5/3} \).

So, simplifying and putting it all together, the derivative of the function \( f(x) \) is given by:

To use LaTeX correctly to write this, we would use:

And there you have it, the derivative of the function, which we can use to find the slope of the tangent line to the curve at any particular \( x \) value.