Find y' for y = x^4 - a^n, where a and n are constants
Find $y'$ for $y = x^{4} - a^{n}$, where $a$ and $n$ are constants.
Explanation of Method
Recognise which symbols are variables and which are constants.
Step 1: Identify each term
The function has two terms. The first, $x^{4}$, is a power of the independent variable $x$. The second, $a^{n}$, is built only from the constants $a$ and $n$, so it is itself a constant and does not depend on $x$ at all.
Step 2: Differentiate term by term
\[ y' = \frac{d}{dx}\bigl(x^{4}\bigr) - \frac{d}{dx}\bigl(a^{n}\bigr). \]
By the power rule $(x^{n})' = nx^{n-1}$ and the constant rule $(c)' = 0$: $\frac{d}{dx}\bigl(x^{4}\bigr) = 4x^{3}, \qquad \frac{d}{dx}\bigl(a^{n}\bigr) = 0$.
Result
Final answer:
\[\boxed{\,y' = 4x^{3}\,}\]
Remark
Remark. The point of this exercise is notational vigilance. The expression $a^{n}$ looks like a power function, but the variable $x$ does not occur in it, so it differentiates to zero. Compare Problem 18, where the variable sits in the exponent and $\bigl(a^{x}\bigr)' = a^{x}\ln a \neq 0$.