Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please only solve using the given formulas as it'll help me more
(PQ1)
Transcribed Image Text:1. Each of the following limit expressions correspond to a limit definition of the derivative of a particular function at a
specific point. Determine the function and the point for each expression.
5 sin(20) – 5
lim
((2+ h)³ + 2(2 + h)² – 9) – 7
а.
b. lim
h-0
h
Function
Point
Function
Point
3
2
3
2
![DEFINITIONS OF THE DERIVATIVE
f(x) – f(a)
f(x + h) – f(x).
f(a + h) – f(a)
S'(x) = lim
h-0
S'(a) = lim
S'(a) = lim
h-0
h
X - a
h
MULTIPLE DERIVATIVES
BASIC FUNCTION DERIVATIVES
A function fis n-times differentiable (or f E C") if you can apply
the derivative n times to f and have a continuous function after
cach application of the derivative.
Constant:
d
- [a] = 0, where a ER
dx
A function f is smooth (or f E C®) if it can be differentiated
infinitely many times, and each derivative is a continuous
function.
Power:
r] =r x-1, where r e R
dx
Exponential:
DERIVATIVE PROPERTIES
- b] = In(b) · b*, where b e (0,00)
Lincarity:
dx
Logarithmic:
[Fa) + a · g(x)] =f"(x) + a ·• g'(x)
[log,(x)]:
1
where b, x € (0,00)
Products:
dx
In(b) · x
Trigonometric:
[Scx)g(x)] = f°(x)g (x) + f(x)g°(x)
dx
: [sin(x)]
= cos(x)
Quotients:
dx
f'(x)g (x) – f (x)g'(x)
(8(2)°
d
d
[cos(x)] = – sin(x)
dx
dx g(x
[tan(x)] = sec²(x)
dx
where g (x) # 0
Compositions:
d
[sec(x)]
dx
= sec(x)tan(x)
(8(x)] =S" (8(x)) · g'(x)
d
[cot(x)] = – csc²(x)
dx
d
TANGENT AND NORMAL LINES
- [csc(x)] = - csc(x)cot(x)
dx
If y = f(x) describes some differentiable function, the equation of
the tangent line at a point x = a is given by
Inverse Trigonometric:
d
- (arcsin(x)] =
where x #±1
y = f'(x)(x – a) + f (a).
dx
The equation of the normal line at a point x = a is given by
d
- [arccos(x)] =
where x #+1
1
-(x - a) +f(a).
f'(a)
dx
y = -
d
[arctan(x)] :
dx
1+x2
LINEAR APPROXIMATION
Нурerbolic:
d
[sinh(x)] = cosh(x)
dx
If fis differentiable near x = a, then for values close to a,
f(x) z f'(a)(x – a) +f(a).
d
- [cosh(x)]
= sinh(x)
dx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe24ff362-ce98-42b1-9f6c-651d78be6484%2Ff528154f-f16e-4d89-a86b-f4408a5328e5%2F2uu5qbd_processed.png&w=3840&q=75)
Transcribed Image Text:DEFINITIONS OF THE DERIVATIVE
f(x) – f(a)
f(x + h) – f(x).
f(a + h) – f(a)
S'(x) = lim
h-0
S'(a) = lim
S'(a) = lim
h-0
h
X - a
h
MULTIPLE DERIVATIVES
BASIC FUNCTION DERIVATIVES
A function fis n-times differentiable (or f E C") if you can apply
the derivative n times to f and have a continuous function after
cach application of the derivative.
Constant:
d
- [a] = 0, where a ER
dx
A function f is smooth (or f E C®) if it can be differentiated
infinitely many times, and each derivative is a continuous
function.
Power:
r] =r x-1, where r e R
dx
Exponential:
DERIVATIVE PROPERTIES
- b] = In(b) · b*, where b e (0,00)
Lincarity:
dx
Logarithmic:
[Fa) + a · g(x)] =f"(x) + a ·• g'(x)
[log,(x)]:
1
where b, x € (0,00)
Products:
dx
In(b) · x
Trigonometric:
[Scx)g(x)] = f°(x)g (x) + f(x)g°(x)
dx
: [sin(x)]
= cos(x)
Quotients:
dx
f'(x)g (x) – f (x)g'(x)
(8(2)°
d
d
[cos(x)] = – sin(x)
dx
dx g(x
[tan(x)] = sec²(x)
dx
where g (x) # 0
Compositions:
d
[sec(x)]
dx
= sec(x)tan(x)
(8(x)] =S" (8(x)) · g'(x)
d
[cot(x)] = – csc²(x)
dx
d
TANGENT AND NORMAL LINES
- [csc(x)] = - csc(x)cot(x)
dx
If y = f(x) describes some differentiable function, the equation of
the tangent line at a point x = a is given by
Inverse Trigonometric:
d
- (arcsin(x)] =
where x #±1
y = f'(x)(x – a) + f (a).
dx
The equation of the normal line at a point x = a is given by
d
- [arccos(x)] =
where x #+1
1
-(x - a) +f(a).
f'(a)
dx
y = -
d
[arctan(x)] :
dx
1+x2
LINEAR APPROXIMATION
Нурerbolic:
d
[sinh(x)] = cosh(x)
dx
If fis differentiable near x = a, then for values close to a,
f(x) z f'(a)(x – a) +f(a).
d
- [cosh(x)]
= sinh(x)
dx
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