Chapter 3 – Derivatives of radical expressions
𝑎
𝑏
𝑥 −𝑐 =
𝑥𝑏 = 𝑥𝑎
1
𝑥𝑐
1.) Write as a fractional exponent:
a)
𝑥3
b)
4
𝑥3
3
𝑐) 𝑥 5
d)
7
𝑥
2.) Write as a fractional exponent and simplify the term so that it is not a fraction:
𝑎)
1
𝑥7
3
b)
1+𝑥 4
𝑐)
1
d)
5𝑥 3 𝑥 2 −1
6𝑥
4
( 3𝑥 2 )
𝑥 2 +𝑥−7
3.) Write as a radical:
7
a) 𝑥 2
2
b) 𝑥 5
3
5
c) 𝑥 − 4
d) 𝑥 − 3
4.) Write fractional exponents as a radical and make all integer exponents positive:
1
a) 5𝑥 −3 (𝑥 2 + 𝑥 − 3)2
1
b) 3𝑥 2 (𝑥 4 − 1)−2
c) 2𝑥
−2
1
(1 + 𝑥 4 )−2 6−1 ( 𝑥 )
5.) Find the derivative of each of the following functions:
5
2
a)
f (x) = 𝑥 3 − 𝑥 3
b)
f (t) =
c)
f (h) = 2 ℎ + 3ℎ
d)
f (c) = c 𝑐 +
e)
f (x) =
f)
f (b) =
g)
f (x) =
5
𝑡 + 4 𝑡5
5
𝑐2
𝑥 2 +4𝑥+3
𝑥
𝑏+𝑏
𝑏2
𝑥 +
1
3
𝑥
2
Answer key
𝑎
Chapter 3 – Derivatives of radical expressions
𝑏
𝑥 −𝑐 =
𝑥𝑏 = 𝑥𝑎
1
𝑥𝑐
1.) Write as a fractional exponent:
a)
𝑥3
4
b)
3
3
𝑥3
𝑐) 𝑥 5
5
3
𝑎) 𝑥 2
d)
𝑥
1
c) 𝑥 3
b) 𝑥 4
7
d) 𝑥 7
2.) Write as a fractional exponent and simplify the term so that it is not a fraction:
1
𝑎)
𝑥7
7
𝑎) 𝑥 − 2
3
b)
𝑐)
1+𝑥 4
b) 3 1 + 𝑥 4
1
2
−
1
d)
5𝑥 3 𝑥 2 −1
c ) 5−1 𝑥 −3 𝑥 2 − 1
−
1
2
6𝑥
4
( 3𝑥 2 )
𝑥 2 +𝑥−7
d ) 2𝑥 −1 𝑥 2 + 𝑥 − 7
3.) Write as a radical:
7
a) 𝑥 2
a)
2
𝑥7
2
3
5
b)
5
c) 𝑥 − 4
b) 𝑥 5
c)
𝑥2
d) 𝑥 − 3
1
4
d)
𝑥3
1
3
𝑥5
4.) Write fractional exponents as a radical and make all integer exponents positive:
1
a) 5𝑥 −3 (𝑥 2 + 𝑥 − 3)2
a)
5 𝑥 2 +𝑥 − 3
𝑥3
1
b) 3𝑥 2 (𝑥 4 − 1)−2
b)
3𝑥 2
𝑥4 − 1
c) 2𝑥
c)
−2
1
(1 + 𝑥 4 )−2 6−1 ( 𝑥 )
1
24𝑥 1+ 𝑥 4
−
1
4
Answer key
5.) Find the derivative of each of the following functions:
5
a)
3
5∙ 𝑥 2
f‘(x)=
b)
f (t) =
5
f‘(t)=
c)
2
f (x) = 𝑥 3 − 𝑥 3
3
1
5
5∙
𝑡4
+ 10𝑡 𝑡
f (h) = 2 ℎ + 3ℎ
3
f (x) =
f (b) =
f (x) =
2
+
2
5
5∙ 𝑐 3
𝑥
3 𝑥
2
2
+
𝑥
3
−
2𝑥 𝑥
𝑏+𝑏
𝑏2
f‘(b)=
g)
𝑐2
𝑥 2 +4𝑥+3
f‘(x)=
f)
3 𝑐
2 ℎ
5
f (c) = c 𝑐 +
f‘(c)=
e)
3∙ 3 𝑥
𝑡 + 4 𝑡5
f‘(h)= 2+
d)
2
−
−3
2𝑏2 ∙
𝑥 +
f‘ (x ) =1+
𝑏
1
3
1
6
−
3 𝑥5
1
𝑏2
2
𝑥
−
2
3𝑥 ∙
3
𝑥2