# What is the derivative of y=arcsin(x/7)?

Apr 16, 2018

$\frac{d}{\mathrm{dx}} \arcsin \left(\frac{x}{7}\right) = {\left(\setminus \sqrt{49 - {x}^{2}}\right)}^{- 1}$

#### Explanation:

We want to evaluate $\frac{d}{\mathrm{dx}} \arcsin \left(\frac{x}{7}\right)$.

We can apply the chain rule, that $\left(f \circ g\right) ' \left(x\right) = \left(f ' \circ g\right) \left(x\right) g ' \left(x\right)$, and in our case $f = \arcsin$ and $g \left(x\right) = \frac{1}{7} x$. Therefore:

$\frac{d}{\mathrm{dx}} \arcsin \left(\frac{x}{7}\right) = \setminus \frac{1}{\setminus \sqrt{1 - {x}^{2} / {7}^{2}}} \setminus \cdot \frac{d}{\mathrm{dx}} \frac{x}{7}$

$\frac{d}{\mathrm{dx}} \arcsin \left(\frac{x}{7}\right) = \setminus \frac{\frac{1}{7} \frac{d}{\mathrm{dx}} x}{\setminus \sqrt{1 - {x}^{2} / {7}^{2}}}$

$\frac{d}{\mathrm{dx}} \arcsin \left(\frac{x}{7}\right) = \setminus \frac{1}{7 \setminus \sqrt{1 - {x}^{2} / 49}}$

We can further simplify this:

$\frac{d}{\mathrm{dx}} \arcsin \left(\frac{x}{7}\right) = \setminus \frac{1}{\setminus \sqrt{49 \left(1 - {x}^{2} / 49\right)}}$

$\frac{d}{\mathrm{dx}} \arcsin \left(\frac{x}{7}\right) = {\left(\setminus \sqrt{49 - {x}^{2}}\right)}^{- 1}$