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Find the derivative of $x\sin\left(\frac{1}{x}\right)$

Step-by-step Solution

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Final Answer

$\frac{x\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)}{x}$
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Step-by-step Solution

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Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\sin\left(\frac{1}{x}\right)$

$\frac{d}{dx}\left(x\right)\sin\left(\frac{1}{x}\right)+x\frac{d}{dx}\left(\sin\left(\frac{1}{x}\right)\right)$

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$\frac{d}{dx}\left(x\right)\sin\left(\frac{1}{x}\right)+x\frac{d}{dx}\left(\sin\left(\frac{1}{x}\right)\right)$

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Learn how to solve differential calculus problems step by step online. Find the derivative of xsin(1/x). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x and g=\sin\left(\frac{1}{x}\right). The derivative of the linear function is equal to 1. The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if {f(x) = \sin(x)}, then {f'(x) = \cos(x)\cdot D_x(x)}. Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}.

Final Answer

$\frac{x\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)}{x}$

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Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of xsin(1/x) using the product ruleFind derivative of xsin(1/x) using the quotient ruleFind derivative of xsin(1/x) using logarithmic differentiationFind derivative of xsin(1/x) using the definition

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Function Plot

Plotting: $\frac{x\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)}{x}$

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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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