3.8 - Inverse Derivatives

Inverse basics

• The slope of an inverse function is the reciprocal of the slope of the original.
  • NOT the negative reciprocal.
• Also keep in mind that the inputs and outputs to the function switch as well, as do the domain and range.

Shaq chart (aka marley map??)

• Used to calculate slopes of inverse functions.

• Fill this in when doing derivatives.

Inverse Trig Derivatives

• The derivative of arcsin x is ${\displaystyle d/dx(\arcsin x)=1/\sqrt{1-x²}}$, when $-1<x<1$
• The derivative of arccos x is ${\displaystyle d/dx(\arccos x)=-1/\sqrt{1-x²}}$, when $-1<x<1$
• The derivative of arctan x is ${\displaystyle d/dx(\arctan x)=1/(1+x²)}$, for all $x$ in $R$
• The derivative of arccsc x is ${\displaystyle d/dx(cs{c}^{-1}x)=-1/(|x|\sqrt{x²-1})}$, when $x<-1$ or $x>1$
• The derivative of arcsec x is ${\displaystyle d/dx(se{c}^{-1}x)=1/(|x|\sqrt{x²-1}}$), when $x<-1$ or $x>1$
• The derivative of arccot x is ${\displaystyle d/dx(co{t}^{-1}x)=-1/(1+x²)}$, for all $x$ in $R$

Using right triangles to find inverse derivatives

• You don’t have to memorize the above if you use right triangles to find the derivatives instead.

Example

Find the derivative of $y=\arccos \left(\frac{2x}{5}\right)$.

1. Take the cosine of both sides: $\cos y=\frac{2x}{5}$
2. We begin taking a derivative: ${\displaystyle -\sin y\frac{dy}{dx}=\frac{2}{5}}$
3. Solve for $\frac{dy}{dx}$: ${\displaystyle \frac{dy}{dx}=\frac{2}{-5\sin y}}$
4. We need to find $\sin y$. We draw a right triangle, filling in a non-right angle as $y$, its adjacent side as $2x$, and the hypotenuse as $5$. We use the Pythagorean Theorem to find the opposite side in terms of $x$: $\sqrt{25-4x^2}$
5. Now we can find $\sin y$ by doing opposite/hypotenuse: $\sin y=\frac{\sqrt{25-4x^2}}{5}$
6. Plug in $\sin y$. ${\displaystyle \frac{dy}{dx}=-\frac{2}{\sqrt{25-4x^2}}}$