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Answer
La derivada \( y^{\prime} \) es \( -5 + 24x^{5} \).
Solution
Para encontrar la derivada de \( y \) respecto a \( x \), derivamos término a término:
\[
y = 8 - 5x + 4x^{6}
\]
Calculamos la derivada de cada término:
1. La derivada de \( 8 \) es \( 0 \) (ya que es una constante).
2. La derivada de \( -5x \) es \( -5 \).
3. La derivada de \( 4x^{6} \) es \( 24x^{5} \) (utilizando la regla de la potencia).
Por lo tanto, la derivada \( y^{\prime} \) es:
\[
y^{\prime} = 0 - 5 + 24x^{5} = -5 + 24x^{5}
\]
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Simplify this solution Beyond the Answer
To find the derivative \( y' \) of the given polynomial \( y = 8 - 5x + 4x^6 \), we apply the power rule of differentiation. The constant 8 will disappear, the derivative of \( -5x \) is \( -5 \), and using the power rule, the derivative of \( 4x^6 \) is \( 24x^5 \). So, assembling all parts, we get \( y' = -5 + 24x^5 \). Now you have the derivative \( y' = 24x^5 - 5 \). This derivative tells you the slope of the function at any point \( x \), providing insights into the function’s behavior, such as increasing or decreasing intervals!
