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Find the derivative of $4x\sqrt{x^3y^7}+3xy^2\sqrt{x^3y^3}-7y\sqrt{x^5y^5}+\frac{x}{y}\sqrt{25x^3y^9}$

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Solution

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Basic Derivatives

· Sum Rule for Differentiation
$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)$
$\frac{d}{dx}\left(cx\right)=c\frac{d}{dx}\left(x\right)$
· Product rule for derivatives
$\frac{d}{dx}\left(ab\right)=\frac{d}{dx}\left(a\right)b+a\frac{d}{dx}\left(b\right)$
· Derivative of the linear function
$\frac{d}{dx}\left(x\right)=1$
· Power rule for derivatives
$\frac{d}{dx}\left(x^a\right)=ax^{\left(a-1\right)}$

Function Plot

Plotting: $4\left(\sqrt{x^{3}}\sqrt{y^{7}}+x\left(\frac{3}{2}\sqrt{x}\sqrt{y^{7}}+\frac{7}{2}\sqrt{x^{3}}\sqrt{y^{5}}\right)\right)+3\left(\sqrt{y^{7}}\sqrt{x^{3}}+x\left(2y\sqrt{x^{3}}\sqrt{y^{3}}+y^2\left(\frac{3}{2}\sqrt{x}\sqrt{y^{3}}+\frac{3}{2}\sqrt{x^{3}}\sqrt{y}\right)\right)\right)-7\left(\sqrt{x^{5}}\sqrt{y^{5}}+y\left(\frac{5}{2}\sqrt{x^{3}}\sqrt{y^{5}}+\frac{5}{2}\sqrt{x^{5}}\sqrt{y^{3}}\right)\right)+5\left(\frac{5}{2}\sqrt{x^{3}}\sqrt{y^{7}}+\frac{7}{2}\sqrt{x^{5}}\sqrt{y^{5}}\right)$

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d/dx
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|◻|
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sin
cos
tan
cot
sec
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asin
acos
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acot
asec
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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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