Method: Inverse Function Derivative

Step 1: Simplify and look for algebraic rewrites.

This is your new version of Option 1 from limits. This means it is the first thing you will always want to do before you start actually applying a derivative rule. You must ensure that the equation is ready to have a derivative taken.

• Roots or Radicals
• x’s on the bottom of a fraction.
• Trig functions raised to a power.

Step 2: Make sure you have both the original function, $f\mathrm{(}x\mathrm{)}$, and the inverse function, $f^{–1}\left(x\right)$.

• If you missing one of the equations, you can find the other through an algebra.
• No matter which one your given to start with, you can find the other through this process.
  • Swap the position of x and y in the equation you are given. Keep in mind that $f\mathrm{(}x\mathrm{)}$ or $f^{–1}\left(x\right)$ are the y .
  • Resolve that new equation for y . In other words, get y alone on one side of the equals. That result is your other equation.

Step 3: Find the derivative of $f\mathrm{(}x\mathrm{)}$, the original equation.

Step 4: Construct the $f\prime \mathrm{(}{f}^{–1}\left(x\right)\mathrm{)}$ piece of the inverse derivative rule.

Step 5: Bring the pieces together following the inverse derivative rule, $\mathrm{(}{f}^{–1}\mathrm{)}\prime \left(x\right)=\frac{1}{f\prime \mathrm{(}{f}^{–1}\left(x\right)\mathrm{)}}$.