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Find the derivative of $\arcsin\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)$

Step-by-step Solution

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Final Answer

$\frac{\frac{\sin\left(x^2\right)}{2\sqrt{x}}+2\left(-\sqrt{x}-3\right)x\cos\left(x^2\right)}{\sqrt{-x-6\sqrt{x}-9+\sin\left(x^2\right)^2}\sin\left(x^2\right)}$
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Step-by-step Solution

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Taking the derivative of arcsine

$\frac{1}{\sqrt{1-\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)^2}}\frac{d}{dx}\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)$

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$\frac{1}{\sqrt{1-\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)^2}}\frac{d}{dx}\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)$

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Learn how to solve problems step by step online. Find the derivative of arcsin((x^1/2+3)/sin(x^2)). Taking the derivative of arcsine. The power of a quotient is equal to the quotient of the power of the numerator and denominator: \displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}. Multiplying the fraction by -1. 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

$\frac{\frac{\sin\left(x^2\right)}{2\sqrt{x}}+2\left(-\sqrt{x}-3\right)x\cos\left(x^2\right)}{\sqrt{-x-6\sqrt{x}-9+\sin\left(x^2\right)^2}\sin\left(x^2\right)}$

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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 arcsin((x^0.5+3)/sin(x^2)) using the product ruleFind derivative of arcsin((x^0.5+3)/sin(x^2)) using the quotient ruleFind derivative of arcsin((x^0.5+3)/sin(x^2)) using logarithmic differentiation

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Function Plot

Plotting: $\frac{\frac{\sin\left(x^2\right)}{2\sqrt{x}}+2\left(-\sqrt{x}-3\right)x\cos\left(x^2\right)}{\sqrt{-x-6\sqrt{x}-9+\sin\left(x^2\right)^2}\sin\left(x^2\right)}$

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7
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9
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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