Practice Problems on Differentiation : Hints
Math 1220(N.Mackey)
In each case, find an expression for the derivative.
11
1. y = √ 2
2 x −x+9
Treat
11
2
as a constant. Then use the laws of exponents to rewrite the expression involving x.
√
w
−1
Use the quotient rule.
2. y =
w3
√
3. y =
w
3
w −1
!7
One can use the chain rule. Another nice option is to use logarithmic differentiation. That is,
take ln of both sides and simplify using laws of logs. Differentiation is now a breeze. Finally,
use the original equation to substitute for y.
4. y = ex + xe + e
The second term is a power function, the third is a constant.
θ2 + 1
5. y = √
3
θ2 + 3
One option is to use the quotient rule. Another is to use logarithmic differentiation.
−1
6. y = etan
x
(x2 + 1)
Start with the product rule.
7. y = cos3 (5x2 )
Rewrite cos3 (5x2 ) as (cos(5x2 ))3 .
8. sin(xy) = x2 − y
Use implicit differentiation.
9. y = e2x sec(2x) tan(2x)
Use the product rule. Recall (f gh)0 = f 0 gh + f g 0 h + f gh0 .
10. x2 (x − y)2 = x2 − y 2
Use implicit differentiation. The product rule is used for the left hand side.
11. y =
v
u 2
u
(x
5

ln t

+ 1)13 
(2 − x3 )20
First simplify using laws of logs. Then differentiation is easy!
12. y =
v
u
u
x(x + 2)3
3
t
(1 − x2 )7
Use logarithmic differentiation. That is, take ln of both sides and simplify using laws of logs.
Differentiation is now a breeze. Finally, use the original equation to substitute for y.
13. y =
√
x ex
2 −1
sin−1 (2x)
Begin with the product rule.
t3 + 5
√
t t
You can directly use the quotient rule, but some preliminary simplification would be advantageous. Rewrite the denominator using the laws of exponents and then distribute the
denominator. Use laws of exponents once again to rewrite each of the two terms. Differentiation will now be straigtforward.
14. y =
15. y =
5(e2t + e−2t )
3(e2t − e−2t )
Treat 35 as a constant and begin with the constant multiple rule, that is (cf )0 = c · f 0 .
Before differentiation, you might want to simplify the quotient by multiplying numerator
and denominator by e2t . Then use the quotient rule. The chain rule will be needed when
differentiating the numerator as well as the denominator. A nice simplification is possible
after differentiation is complete.
16. tan(x3 y 2 ) = 1
Use implicit differentiation.
Last Updated: Monday 26th October, 2015,
19:02.
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