A3.4

verify that the power rule applies to functions of the form $f(x)=x^n$, where $n$ is a rational number [e.g., by comparing values of the slopes of tangents to the function $f(x)=x^{1/2}$ with values of the derivative function determined using the power rule], and verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for $f(x)=(5x^3{)}^{1/3}$ by using the chain rule and by differentiating the simplified form, $f(x)=5^{1/3}x$] and the product rule using polynomial functions [e.g., by determining the same derivative for $f(x)=(3x+2)(2x^2-1)$ by using the product rule and by differentiating the expanded form $f(x)=6x^3+4x^2-3x-2$] Sample problem: Verify the chain rule by using the product rule to look for patterns in the derivatives of $f(x)=x^2+1$, $f(x)=(x^2+1{)}^2$, $f(x)=(x^2+1{)}^3$, and $f(x)=(x^2+1{)}^4$.