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Find the derivative of $\sqrt{\frac{x\left(2x+3\right)^5}{\left(7x-10\right)^{15}}}$

Step-by-step Solution

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Final Answer

$\frac{\left(\frac{\sqrt{\left(2x+3\right)^{5}}}{2\sqrt{x}}+5\sqrt{x}\sqrt{\left(2x+3\right)^{3}}\right)\sqrt{\left(7x-10\right)^{15}}-\frac{105}{2}\sqrt{x}\sqrt{\left(2x+3\right)^{5}}\sqrt{\left(7x-10\right)^{13}}}{\left(7x-10\right)^{15}}$
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Step-by-step Solution

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The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\frac{d}{dx}\left(\frac{\sqrt{x\left(2x+3\right)^5}}{\sqrt{\left(7x-10\right)^{15}}}\right)$

Learn how to solve differential calculus problems step by step online.

$\frac{d}{dx}\left(\frac{\sqrt{x\left(2x+3\right)^5}}{\sqrt{\left(7x-10\right)^{15}}}\right)$

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Learn how to solve differential calculus problems step by step online. Find the derivative of ((x(2x+3)^5)/((7x-10)^15))^1/2. 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}. The power of a product is equal to the product of it's factors raised to the same power. 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}}. Simplify \left(\sqrt{\left(7x-10\right)^{15}}\right)^2 using the power of a power property: \left(a^m\right)^n=a^{m\cdot n}. In the expression, m equals \frac{15}{2} and n equals 2.

Final Answer

$\frac{\left(\frac{\sqrt{\left(2x+3\right)^{5}}}{2\sqrt{x}}+5\sqrt{x}\sqrt{\left(2x+3\right)^{3}}\right)\sqrt{\left(7x-10\right)^{15}}-\frac{105}{2}\sqrt{x}\sqrt{\left(2x+3\right)^{5}}\sqrt{\left(7x-10\right)^{13}}}{\left(7x-10\right)^{15}}$

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Function Plot

Plotting: $\frac{\left(\frac{\sqrt{\left(2x+3\right)^{5}}}{2\sqrt{x}}+5\sqrt{x}\sqrt{\left(2x+3\right)^{3}}\right)\sqrt{\left(7x-10\right)^{15}}-\frac{105}{2}\sqrt{x}\sqrt{\left(2x+3\right)^{5}}\sqrt{\left(7x-10\right)^{13}}}{\left(7x-10\right)^{15}}$

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7
8
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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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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