Video Transcript
Evaluate the derivative of the
inverse sin of the square root of one minus 𝑥 squared with respect to 𝑥.
Here, we have a function of a
function or a composite function. So we’ll use the chain rule to
find its derivative. This says that if 𝑦 is some
function in 𝑢 and 𝑢 is some function in 𝑥, then d𝑦 by d𝑥 is equal to d𝑦 by
d𝑢 times d𝑢 by d𝑥. We’ll let 𝑢 be equal to the
square root of one minus 𝑥 squared. Which can, of course,
alternatively be written as one minus 𝑥 squared to the power of one-half. Then 𝑦 is equal to the inverse
sin of 𝑢. To apply the chain rule, we’re
going to need to find the derivative of both of these functions. The derivative of the inverse
sin of 𝑢 with respect to 𝑢 is one over the square root of one minus 𝑢
squared.
And we can use the general
power rule to find the derivative of one minus 𝑥 squared to the power of
one-half. It’s a half times one minus 𝑥
squared to the negative one-half times the derivative of the bit inside the
brackets which is negative two 𝑥. That can be written as negative
𝑥 times one minus 𝑥 squared to the power of negative one-half.
d𝑦 by d𝑥 is, therefore,
negative 𝑥 over the square root of one minus 𝑥 squared times one over the
square root of one minus 𝑢 squared. We can replace 𝑢 with one
minus 𝑥 squared to the power of one-half. And the second fraction becomes
one over the square root of one minus one minus 𝑥 squared. This further simplifies to one
over 𝑥. And we divide through by
𝑥. And we see that the derivative
of our function is negative one over the square root of one minus 𝑥
squared.
