Preview Activity 5.3.1. In Section 2.5, we learned the Chain Rule and how it can be applied to find the derivative of a composite function. In particular, if u is a differentiable function of x, and f is a differentiable function of u(x), then −[ƒ(u(x))] = fƒ'(u(x)) · u'(x). d dx In words, we say that the derivative of a composite function c(x) = f(u(x)), where f is considered the "outer" function and u the "inner" function, is "the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function." a. For each of the following functions, use the Chain Rule to find the function's derivative. Be sure to label each derivative by name (e.g., the derivative of g(x) should be labeled g'(x)). 3x i. g(x) = e³ iii. p(x) = arctan(2x) V. r(x) = 34-11x b. For each of the following functions, use your work in (a) to help you determine the general antiderivative of the function. Label each antiderivative by name (e.g., the antiderivative of m should be called M). In addition, check your work by computing the derivative of each proposed antiderivative. iii. s(x) = - ii. h(x) = sin(5x + 1) iv. g(x) = (2-7x) ª i. m(x) = e³x 1 1+ 4x² V. w(x) = 34-11x c. Based on your experience in parts (a) and (b), conjecture an antiderivative ii. n(x) = cos(5x + 1) iv. v(x) = (2 - 7x)³
Preview Activity 5.3.1. In Section 2.5, we learned the Chain Rule and how it can be applied to find the derivative of a composite function. In particular, if u is a differentiable function of x, and f is a differentiable function of u(x), then −[ƒ(u(x))] = fƒ'(u(x)) · u'(x). d dx In words, we say that the derivative of a composite function c(x) = f(u(x)), where f is considered the "outer" function and u the "inner" function, is "the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function." a. For each of the following functions, use the Chain Rule to find the function's derivative. Be sure to label each derivative by name (e.g., the derivative of g(x) should be labeled g'(x)). 3x i. g(x) = e³ iii. p(x) = arctan(2x) V. r(x) = 34-11x b. For each of the following functions, use your work in (a) to help you determine the general antiderivative of the function. Label each antiderivative by name (e.g., the antiderivative of m should be called M). In addition, check your work by computing the derivative of each proposed antiderivative. iii. s(x) = - ii. h(x) = sin(5x + 1) iv. g(x) = (2-7x) ª i. m(x) = e³x 1 1+ 4x² V. w(x) = 34-11x c. Based on your experience in parts (a) and (b), conjecture an antiderivative ii. n(x) = cos(5x + 1) iv. v(x) = (2 - 7x)³
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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only need help with questions a and b, please.
thank you!
![Preview Activity 5.3.1. In Section 2.5, we learned the Chain Rule and how it
can be applied to find the derivative of a composite function. In particular, if u is
a differentiable function of x, and f is a differentiable function of u(x), then
−[ƒ(u(x))] = f'(u(x)) · u'(x).
น
In words, we say that the derivative of a composite function c(x) = f(u(x)),
where f is considered the "outer" function and u the “inner" function, is "the
derivative of the outer function, evaluated at the inner function, times the
derivative of the inner function."
a. For each of the following functions, use the Chain Rule to find the
function's derivative. Be sure to label each derivative by name (e.g., the
derivative of g(x) should be labeled g'(x)).
i. g(x) = e³x
3x
dx
3x
i. m(x) = e³x
İİİ. p(x) = arctan(2x)
V. r(x) = 34-11x
b. For each of the following functions, use your work in (a) to help you
determ the general antiderivative of the function. Label each
antiderivative by name (e.g., the antiderivative of m should be called M).
In addition, check your work by computing the derivative of each
proposed antiderivative.
ii. h(x) = sin(5x + 1)
iv. q(x) = (2 – 7x)ª
iii. s(x) =
=
ii. n(x) = cos(5x + 1)
iv. v(x) = (2 — 7x)³
1
1+4x²
V. w(x) = 34-11x
c. Based on your experience in parts (a) and (b), conjecture an antiderivative](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce3589f0-5608-4fb5-a11c-ce1805f2c8e6%2F05391c17-257a-40e9-a0e4-2809c46ab00f%2Fzf4qt5g_processed.png&w=3840&q=75)
Transcribed Image Text:Preview Activity 5.3.1. In Section 2.5, we learned the Chain Rule and how it
can be applied to find the derivative of a composite function. In particular, if u is
a differentiable function of x, and f is a differentiable function of u(x), then
−[ƒ(u(x))] = f'(u(x)) · u'(x).
น
In words, we say that the derivative of a composite function c(x) = f(u(x)),
where f is considered the "outer" function and u the “inner" function, is "the
derivative of the outer function, evaluated at the inner function, times the
derivative of the inner function."
a. For each of the following functions, use the Chain Rule to find the
function's derivative. Be sure to label each derivative by name (e.g., the
derivative of g(x) should be labeled g'(x)).
i. g(x) = e³x
3x
dx
3x
i. m(x) = e³x
İİİ. p(x) = arctan(2x)
V. r(x) = 34-11x
b. For each of the following functions, use your work in (a) to help you
determ the general antiderivative of the function. Label each
antiderivative by name (e.g., the antiderivative of m should be called M).
In addition, check your work by computing the derivative of each
proposed antiderivative.
ii. h(x) = sin(5x + 1)
iv. q(x) = (2 – 7x)ª
iii. s(x) =
=
ii. n(x) = cos(5x + 1)
iv. v(x) = (2 — 7x)³
1
1+4x²
V. w(x) = 34-11x
c. Based on your experience in parts (a) and (b), conjecture an antiderivative
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