Question

find \\(\frac{dy}{du}\\), \\(\frac{du}{dx}\\), and \\(\frac{dy}{dx}\\). \\(y = u^{55}\\) and \\(u = 2x^3 - 4x^2\\) \\(\frac{dy}{du} = 55u^{54}\\) \\(\frac{du}{dx} = \square\\)

Explanation:

Step1: Differentiate \( u = 2x^3 - 4x^2 \)

Using the power rule \( \frac{d}{dx}(x^n) = nx^{n - 1} \), we differentiate each term.
For \( 2x^3 \), the derivative is \( 2\times3x^{3 - 1} = 6x^2 \).
For \( -4x^2 \), the derivative is \( -4\times2x^{2 - 1} = -8x \).

Step2: Combine the derivatives

Adding the derivatives of the two terms, we get \( \frac{du}{dx} = 6x^2 - 8x \).

Answer:

\( 6x^2 - 8x \)