Inverse Derivative Problems
1. Let f be the function defined by f(x) = x3 + x. If g(x) = f –1 (x) and g(2) = 1,
what is the value of g ′(2)?
2. Let f be the function defined by f(x) =
4+𝑥 5
3𝑥 2
. If g(x) = f –1 (x), and g(3)=2, what
is the value of g ′(3)?
3. Let g(x) = f –1 (x) and f(1)=6, f’(1) = 7, what is the value of g ′(6)?
4. Let g(x) = f –1 (x) and f(2)=10, f’(2) = 5, what is the value of g ′(10)?
5. Let g(x) = f –1 (x) and f(4)=2, g’(2) = 50, what is the value of f ′(4)?
Inverse Derivative Problems - key
f and g are inverse functions.
f(a) = b
g(b) = a The point (a,b) satisfies f. The point (b,a) satisfies g.
Inverse Derivative Formula:
𝑓 ‘ (a) =
1
g ′(b)
g ‘ (b) =
and
1
f ′(a)
1. Let f be the function defined by f(x) = x3 + x. If g(x) = f –1 (x) and g(2) = 1,
what is the value of g ′(2)?
Not possible to find inverse function directly. x = y3+y cannot be solved for y. So use inverse
derivative formula. The point (1,2) satisfies f. the point (2,1) satisfies g. g ‘ (2) =
1
f ′(1)
We need f ’(1).
f’ (x) = 3x2+1 f ‘(1) = 3 (1)2+1= 4
g ‘ (2) =
1
f ′(1)
=
2. Let f be the function defined by f(x) =
1
4
4+𝑥 5
3𝑥 2
. If g(x) = f –1 (x), and g(3)=2, what
is the value of g ′(3)?
3 is the b value. The point (2,3) satisfies f. the point (3,2) satisfies g. using output b=3, we can also find
a if not given. Output from f(x) = 3, what was x? 3 =
4+𝑥 5
3𝑥 2
9x2= 4+x5 x5 - 9x2 +4 = 0
From graph or guessing, we could find x = 2 (but there are other values as well, so good that they gave
us this value). g(3) = 2. So f(2) = 3.
Not possible to find inverse function directly. x =
3x2 (5x4 )−(4+x5 )(6x)
9𝑥 4
cannot be solved for y. So use inverse
1
f ′(2)
derivative formula. g ‘ (3) =
f’ (x) =
4+𝑦 5
3𝑦
f ‘(2) = (960-432)/(9*16)=528/144 = 11/3 g ‘ (3) =
1
f ′(2)
=
1
11
3
3. Let g(x) = f –1 (x) and f(1)=6, f’(1) = 7, what is the value of g ′(6)?
g ‘ (6) =
1
1
=
f ′(1)
7
4. Let g(x) = f –1 (x) and f(2)=10, f’(2) = 5, what is the value of g ′(10)?
g ‘ (10) =
1
1
=
f ′(2)
5
5. Let g(x) = f –1 (x) and f(4)=2, g’(2) = 50, what is the value of f ′(4)?
f ‘ (4) =
1
1
=
g ′(2)
50
3
= 11