Exam (elaborations)
The Chain Rule solved questions
- Course
- Institution
The Chain Rule solved questions
[Show more]
Seller
Follow
jureloqoo
Content preview
CHAPTER 9The Chain Rule
9.1 If f(x) = x2 + 2x — 5 and g(x) = x3, find formulas for the composite functions /°g and g°/.
(/°g)W=/(g(*))=/(* 3 ) = (*3)2 + 2(* 3 )-5 = * 6 +2* 3 -5
(g°/)« = S(/M) = 8(*2 + 2x-5) = (x2 + 2x- 5)3
9.2 Write the function is the composition of two functions.
Let g(x) = 3x - 5 and let f(x) = Vx. Then ( f°*)<*) = /(gW) = A3* - 5) = V3l^5.
9.3 If f(x) = 2x and g(x) = \/(x- 1), find all solutions of the equation (f°g)(x) = (g°/)(x).
(/•*)(*)= A *(*))=/ and (*•/)(*) = *( A*)) = *(2*) =
So, we must solve 4x-2 = x-l, 3x = l, * = 5 . Answer
9.4 Write the chain rule formula for the derivative of f» g
(/"*)'(*) =/'(*<*))•*'(*)•
9.5 If y = F(u) and M = G(x), then we can write y = F(G(x)). Write the chain rule formula for dyldx,
where we think of y as a function of x.
Here, the first occurrence of y refers to y as a function of AC, while the second occurrence of y
(on the right side) refers to y as a function of «.
9.6 Find the derivative of (*3 - 2x2 +7x- 3)4.
Use the chain rule. Think of the function as a composition (f°g)(x), where f(x) = x4 and g(x) =
x3 - 2x2 + 7x-3. Then f '(x) = 4x3 and g'(jr) = 3* -4x + 7. Hence, x3 - 2x2 + 7x- 3)4 = 4(x3 -
2 3 2
2x + 7x- 3) • (3x -4x + 7).
9.7 Find the derivative of
We can write = (3*2 + 5)-4. Now we can use the chain rule. Remember that x'^rr'- 1
for any real number r. In particular, »-« = -4x-5. By the chain rule,
(3x2 + 5)"4 = -4(3x2 + 5K5 • (6jr) = -
9.8 Find the derivative of
We can write = (2x + 7)1'2. By the chain rule,
Here, we have used and
9.9 Find the derivative of
56
, THE CHAIN RULE 57
Use the chain rule. Here, we must calculate by
the quotient rule: Hence,
Answer
9.10 Find the derivative of (4x2 - 3)2(x + 5)3.
Think of this function as a product of (4x2 - 3)2 and (x + 5)3, and first apply the product rule:
[(4x2-3)2(jt + 5)3] = (4;c 2 -3) 2 - (x + 5)3 + (x + 5)3 • (4*2-3)2. By the chain rule, (* + 5)3 =
2 2 2 2 2 2
3(* + 5) • 1 = 3(* + 5) , jnd (4x - 3) = 2(4* - 3) • (8x) = I6x(4x - 3). Thus, t(4^ 2 -3) 2 (jc +
3 2 2 2 3 2 2
5) ] = (4^ - 3) • 3(x + 5) + (x + 5) -16^(4jc -3). We can factor out (4x - 3) and (x + 5)2 to ob-
tain: (4^;2 - 3)(x + 5)2[3(4*2 - 3) +16^ + 5)] = (4x2 - 3)(x + 5f(12x2 - 9 + 16x2 + 80x) = (4x 2 -3)(* +
5)2(28x2 + 80A:-9). Answer
9.11 Find the derivative of
The chain rule is unnecessary here. So,
Also, So, Thus,
9.12 Find the derivative of
By the chain rule, But,
Hence,
9.13 Find the slope-intercept equation of the tangent line to the graph of at the point (2, 5)-
By the quotient rule,
By the chain rule,
Thus,
When x = 2, and, therefore, at the point (2, 5), Hence, a point-slope equation of
the tangent line is Solving tor y, we obtain the slope-intercept equation
9.14 Find the slope-intercept equation of the normal line to the curve it the point (3,5).
2 2
3, = (* + 16)" . Hence, by the chain rule,
At the point (3,5), and, therefore. This is the slope of the tangent line. Hence, the
slope of the normal line is — § , and a point-slope equation for it is Solving for y, we obtair
the slope-intercept equation
9.15 If y = x —2 and x = 3z2 + 5, then y can be considered a function of z. Express in terms of z.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jureloqoo. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.04. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67474 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now