136
2.6. DERIVATIVES OF INVERSE FUNCTIONS
(f) Use the results of your work above to find an expression involving only 1 and x
for r ′ (x).
⊳
While derivatives for other inverse trigonometric functions can be established similarly,
we primarily limit ourselves to the arcsine and arctangent functions. With these rules
added to our library of derivatives of basic functions, we can differentiate even more
functions using derivative shortcuts. In Activity 2.18, we see each of these rules at work.
Activity 2.18.
Determine the derivative of each of the following functions.
(a) f (x) = x
3 arctan(x) + e
x ln(x)
(b) p(t) = 2
t arcsin(t)
(c) h(z) = (arcsin(5z) + arctan(4 − z))
27
(d) s(y) = cot(arctan(y))
(e) m(v) = ln(sin
2 (v) + 1)
(f) g(w) = arctan
ln(w)
1 + w 2
⊳
The link between the derivative of a function and the derivative of its inverse
In Figure 2.9, we saw an interesting relationship between the slopes of tangent lines to
the natural exponential and natural logarithm functions at points that corresponded to
reflection across the line y = x. In particular, we observed that for a point such as (ln(2), 2)
on the graph of f (x) = e x , the slope of the tangent line at this point is f ′ (ln(2)) = 2,
while at the corresponding point (2, ln(2)) on the graph of f −1 (x) = ln(x), the slope of the
tangent line at this point is ( f −1 ) ′ (2) =
1
2 , which is the reciprocal of f ′ (ln(2)).
That the two corresponding tangent lines having slopes that are reciprocals of one
another is not a coincidence. If we consider the general setting of a differentiable function
f with differentiable inverse g such that y = f (x) if and only if x = g(y), then we know
that f (g(x)) = x for every x in the domain of f −1 . Differentiating both sides of this
equation with respect to x, we have
d
dx
[ f (g(x))] =
d
dx
[x],
and by the chain rule,
f
′ (g(x))g
′ (x) = 1.
2.6. DERIVATIVES OF INVERSE FUNCTIONS
(f) Use the results of your work above to find an expression involving only 1 and x
for r ′ (x).
⊳
While derivatives for other inverse trigonometric functions can be established similarly,
we primarily limit ourselves to the arcsine and arctangent functions. With these rules
added to our library of derivatives of basic functions, we can differentiate even more
functions using derivative shortcuts. In Activity 2.18, we see each of these rules at work.
Activity 2.18.
Determine the derivative of each of the following functions.
(a) f (x) = x
3 arctan(x) + e
x ln(x)
(b) p(t) = 2
t arcsin(t)
(c) h(z) = (arcsin(5z) + arctan(4 − z))
27
(d) s(y) = cot(arctan(y))
(e) m(v) = ln(sin
2 (v) + 1)
(f) g(w) = arctan
ln(w)
1 + w 2
⊳
The link between the derivative of a function and the derivative of its inverse
In Figure 2.9, we saw an interesting relationship between the slopes of tangent lines to
the natural exponential and natural logarithm functions at points that corresponded to
reflection across the line y = x. In particular, we observed that for a point such as (ln(2), 2)
on the graph of f (x) = e x , the slope of the tangent line at this point is f ′ (ln(2)) = 2,
while at the corresponding point (2, ln(2)) on the graph of f −1 (x) = ln(x), the slope of the
tangent line at this point is ( f −1 ) ′ (2) =
1
2 , which is the reciprocal of f ′ (ln(2)).
That the two corresponding tangent lines having slopes that are reciprocals of one
another is not a coincidence. If we consider the general setting of a differentiable function
f with differentiable inverse g such that y = f (x) if and only if x = g(y), then we know
that f (g(x)) = x for every x in the domain of f −1 . Differentiating both sides of this
equation with respect to x, we have
d
dx
[ f (g(x))] =
d
dx
[x],
and by the chain rule,
f
′ (g(x))g
′ (x) = 1.