# Derivatives of Power Functions --- ## What are Power Functions? A power function has the form $f(x) = ax^b$, where $a$ and $b$ are real numbers. --- ## Some Easy Examples of Power Functions: * $p(x) = x^2$ * $q(x) = 5x^4$ * $r(x) = x^{1/3}$ * $y(x) = x^{-2}$ --- ## How do I take the Derivative of a Power Function? **Power Rule:** If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. --- **Power Rule (alternate):** If $f(x) = ax^b$, then $f'(x) = abx^{b-1}$. In words we would say, "if $f(x) = ax^b$, then its derivative is $abx^{b-1}$". --- --- ## Taking Derivatives of Power Functions: ### Example 1 If $p(x) = x^2$, then $p'(x) = 2x$. --- ### Example 2 If $q(x) = 5x^4$, then $q'(x) = 20x^3$. --- ### Example 3 If $r(x) = x^{1/3}$, then $r'(x) = \displaystyle\frac{1}{3}x^{-2/3}$. --- ### Example 4 If $y(x) = x^{-2}$, then $y'(x) = -2x^{-3}$. --- ## Some Trickier Examples of Power Functions Mathematicians have developed some conventions for writing expressions involving powers. Because of these conventions, there are ways of writing power functions that don't immediately fit the easy form we just saw. --- ## :bulb: <span style = "color:red"> Roots and Fractional Powers </span> :key: **When we take the $n$th root of $x$, that is the same as $x^{1/n}$.** If we have a root function, we can rewrite it as a fractional power. --- **Some Examples:** * The cube root of $x$: $C(x) = \sqrt[3]{x}$ can be rewritten as $C(x) = x^{1/3}$. * The square root of $x$: $R(x) = \sqrt{x}$ can be rewritten as $R(x) = x^{1/2}$. --- ## :bulb: <span style = "color:red"> Negative Exponents </span> :key: **$\displaystyle\frac{1}{x^n} = x^{-n}$.** --- **Some Examples:** * $A(x) = \displaystyle\frac{1}{x^3}$ can be rewritten as $x^{-3}$. * $B(x) = \displaystyle\frac{1}{\sqrt{x}}$ can be rewritten as $\displaystyle\frac{1}{x^{1/2}}$, which can be rewritten as $x^{-1/2}$. * If we see $C(x) = 7x^{-4}$, we can know that this is equivalent to $\displaystyle\frac{7}{x^4}$. --- ## :bulb: <span style="color:red"> Product of Powers of a Variable </span> :key: **$x^a x^b = x^{a+b}$.** When we multiply two power functions with the same base, we can add the exponents. --- **Some Examples:** * $\alpha(x) = x^3 x^2$ can be rewritten as $\alpha(x) = x^5$. * If $\omega(x) = 2x^4$ and $\zeta(x) = 3x^{-7}$, then the product $\omega(x)\zeta(x)$ can be rewritten as $2x^4 \cdot 3^{-7} = 6x^{-3}$. --- ## :bulb: <span style = "color:red"> Powers of Powers </span> :key: $(x^a)^b = x^{ab}$ When we raise a power function to another power, we multiply the powers. --- **Some Examples:** * $(x^2)^5 = x^{10}$ * $(7x^4)^{1/2} = 7^{1/2}x^2 = \sqrt{7}x^2$ * $\left( \displaystyle\frac{3}{x^5} \right)^4 = \displaystyle\frac{3^4}{x^{20}} = \displaystyle\frac{81}{x^{20}} = 81x^{-20}$ --- ## More Example Questions for Derivatives --- ### Q1: If $p(x) = 3x^2$, find $p'(x)$. **Answer:** $p'(x) = 3(2)x^{2-1} = 6x$. --- ### Q2: If $q(x) = x^{1/7}$, what is $q'(x)$? **Answer:** $q'(x) = \displaystyle\frac{1}{7}x^{1/7 - 1}$, which we could simplify to $q'(x) = \displaystyle\frac{1}{7}x^{-6/7}$. --- ### Q3: If $D(x) = \sqrt[5]{x^2}$, what is $D'(x)$? **Answer:** First, we rewrite $D(x) = (x^2)^{1/5} = x^{2/5}$, then we take the derivative $D'(x) = \displaystyle\frac{2}{5}x^{-3/5}$. --- ### Q4: If $V(x) = x^4 \cdot 3x^2$, what is $V'(x)$? **Answer:** First, we rewrite $V(x) = 3x^6$. ($x^ax^b = x^{a+b}$) $V'(x) = 3 \cdot 6 x^{6-1}$ $V'(x) = 18x^5$ --- ### Q5: If $D(x) = \left( x^3 \right)^4$, find $D'(x)$. **Answer:** First, we rewrite $D(x) = x^{12}$ $\left( x^a \right)^b = x^{ab})$ Then we take the derivative $D'(x) = 12x^{11}$. ---