Find d by d𝑥 of the inverse
cosecant of 𝑥.
We begin by letting 𝑦 be equal
to the inverse cosecant of 𝑥. And this means we can rewrite
this. And we can say that 𝑥 is equal
to the cosecant of 𝑦.
We’re next going to use
implicit differentiation to find the derivative of both sides of this
equation. The derivative of 𝑥 with
respect to 𝑥 is simply one. Then the derivative of cosec 𝑦
with respect to 𝑥 is equal to the derivative of cosec 𝑦 with respect to 𝑦
times d𝑦 by d𝑥. And the derivative of cosec 𝑦
with respect to 𝑦 is negative cosec 𝑦 cot 𝑦. So we see that one is equal to
negative cosec 𝑦 cot 𝑦 times d𝑦 by d𝑥.
Now, we know that, for the
inverse cosecant function, 𝑦 must be greater than negative 𝜋 by two and less
than 𝜋 by two and not equal to zero. Using these restrictions cosec
𝑦 cot 𝑦 cannot be equal to zero. So we can divide through by
negative cosec 𝑦 cot 𝑦. And we see that d𝑦 by d𝑥 is
as shown. We want to represent our
equation for the derivative in terms of 𝑥. So we’ll use this trigonometric
identity cot squared 𝑦 plus one equals cosec squared 𝑦. And we can rewrite this to say
that cot of 𝑦 is equal to the positive and negative square root of cosec
squared 𝑦 minus one.
We put this into the equation
for the derivative in place of cot of 𝑦. And we then use the fact that
𝑥 is equal to cosec 𝑦. But we are going to need to
make a decision on the sine of the derivative. And it can help here to look at
the graph of the inverse cosecant function. Notice how, for all values of
𝑥 in the range of the function, the derivative of the slope of the tangent is
And we, therefore, use the
absolute value to ensure that our derivative is always negative. We say that d𝑦 by d𝑥 is equal
to the negative of the absolute value of one over 𝑥 times the square root of 𝑥
squared minus one. Since one and the square root
of 𝑥 squared minus one are always positive, we can rewrite this as shown. So the derivative of the
inverse cosecant function 𝑥 with respect to 𝑥 is negative one over the modulus
or absolute value of 𝑥 times the square root of 𝑥 squared minus one.