What is the derivative of \sqrt[4]{sin^3(5x}) ?

The derivative of \sqrt[4]{sin^3(5x}) is (15cos(5x))/(4sin^{1/4)(5x)}

What is the first derivative of \sqrt[4]{sin^3(5x}) ?

The first derivative of \sqrt[4]{sin^3(5x}) is (15cos(5x))/(4sin^{1/4)(5x)}

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\frac{15 cos ( 5 x )}{4 sin ^{\frac{1}{4}} ( 5 x )}

Solution steps

Apply the chain rule:

\frac{1}{4 ( sin ^{3} ( 5 x ) ) ^{\frac{3}{4}}} \frac{d}{d x} (sin^{3} (5 x))

= \frac{1}{4 ( sin ^{3} ( 5 x ) ) ^{\frac{3}{4}}} \frac{d}{d x} (sin^{3} (5 x))

\frac{d}{d x} (sin^{3} (5 x)) = 15 sin^{2} (5 x) cos (5 x)

= \frac{1}{4 ( sin ^{3} ( 5 x ) ) ^{\frac{3}{4}}} \cdot 15 sin^{2} (5 x) cos (5 x)

Simplify

\frac{1}{4 ( sin ^{3} ( 5 x ) ) ^{\frac{3}{4}}} \cdot 15 sin^{2} (5 x) cos (5 x) : \frac{15 cos ( 5 x )}{4 sin ^{\frac{1}{4}} ( 5 x )}

= \frac{15 cos ( 5 x )}{4 sin ^{\frac{1}{4}} ( 5 x )}

What is the derivative of \sqrt[4]{sin^3(5x}) ?
The derivative of \sqrt[4]{sin^3(5x}) is (15cos(5x))/(4sin^{1/4)(5x)}
What is the first derivative of \sqrt[4]{sin^3(5x}) ?
The first derivative of \sqrt[4]{sin^3(5x}) is (15cos(5x))/(4sin^{1/4)(5x)}