Master Calculus! Get instant help on “I Aced Calculus AP” App. Hundreds of flashcards and practice questions at your fingertips. Download now on the App Store and Google Play.

Derivative of $\dfrac3x$

\[\begin{align*} \boxed{\dfrac{d}{dx} \left(\dfrac3x\right) = -\dfrac3{x^2}} \end{align*}\]

Solving for the Derivative

We may pull the constant, $3$, out of the derivative and focus on $\dfrac{d}{dx} \left(\dfrac1x\right)$. This is the same as $x^{-1}$, so we may apply the power rule:

\[\begin{align*} \dfrac{d}{dx} x^n = nx^{n-1} \end{align*}\]

Doing this with $x^{-1}$, we get our derivative is equal to

\[\begin{align*} \dfrac{d}{dx} \left(\dfrac3x\right) &= 3 \dfrac{d}{dx} \left(\dfrac1x\right)\\ &=3 \dfrac{d}{dx} x^{-1}\\ &=3 \cdot -1 \cdot x^{-2}\\ &= -3 \cdot \dfrac{1}{x^2}\\ &=\boxed{-\dfrac{3}{x^2}} \end{align*}\]

The key to this problem is rewriting it in a way that the power rule handles the heavy lifting. It is an important skill to know how to rewrite the problem as a negative exponent when $x$ is in the denominator. We always want positive exponents as the final answer so make sure to reconvert into a fraction at the end!


NEED QUICK

CALC HELP?

Download the I Aced Calculus App today!

ALL Calc Topics, 1000+ of PRACTICE questions


Related Problems

NEED QUICK

  • ALL Calc Topics
    AB and BC
  • 1000+
    PRACTICE questions
  • 400+ FLASHCARDS
  • VIDEO tutorials

CALC HELP?

Download the I Aced Calculus App today!