Find the derivative of y = (1/4)x^12 + 3x^6 - x + 10.

Step-by-step solution

1. 1

   Handle the fractional coefficient carefully. The power rule multiplies the coefficient by the exponent, so the $\frac14$ does not simply carry through: $\left(\frac14 x^{12}\right)^{\prime}=\frac14\cdot 12x^{11}=3x^{11}.$ Writing $\frac{12}{4}=3$ before differentiating the rest avoids the common slip of leaving $\frac14$ in front.

2. 2

   Differentiate the second term. $\left(3x^{6}\right)^{\prime}=3\cdot 6x^{5}=18x^{5}.$

3. 3

   Differentiate the linear and constant terms. A linear term has constant slope, $(-x)^{\prime}=-1$, and any constant has zero derivative, $(10)^{\prime}=0$. The constant $10$ therefore leaves no trace in the answer, which is why $+0$ should not be written in the final expression.

4. 4

   Combine the pieces. $y^{\prime}=3x^{11}+18x^{5}-1 .$

5. 5

   Sanity-check the degrees. Differentiation lowers each degree by exactly one: $12\to 11$, $6\to 5$, $1\to 0$, $0\to$ gone. The result has degree $11$, one less than the original degree $12$, as it must.