What is the derivative of log_{10}(sin(x^2)) ?

The derivative of log_{10}(sin(x^2)) is (2xcot(x^2))/(ln(10))

What is the first derivative of log_{10}(sin(x^2)) ?

The first derivative of log_{10}(sin(x^2)) is (2xcot(x^2))/(ln(10))

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derivative

\frac{2 x cot ( x ^{2} )}{ln ( 10 )}

Solution steps

\frac{d}{d x} (lo g_{10} (sin (x^{2})))

Apply log rule:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{d}{d x} (\frac{ln ( sin ( x ^{2} ) )}{ln ( 10 )})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{ln ( 10 )} \frac{d}{d x} (ln (sin (x^{2})))

Apply the chain rule:

\frac{1}{sin ( x ^{2} )} \frac{d}{d x} (sin (x^{2}))

= \frac{1}{sin ( x ^{2} )} \frac{d}{d x} (sin (x^{2}))

\frac{d}{d x} (sin (x^{2})) = cos (x^{2}) \cdot 2 x

= \frac{1}{ln ( 10 )} \cdot \frac{1}{sin ( x ^{2} )} cos (x^{2}) \cdot 2 x

Simplify

\frac{1}{ln ( 10 )} \cdot \frac{1}{sin ( x ^{2} )} cos (x^{2}) \cdot 2 x : \frac{2 x cot ( x ^{2} )}{ln ( 10 )}

= \frac{2 x cot ( x ^{2} )}{ln ( 10 )}

What is the derivative of log_{10}(sin(x^2)) ?
The derivative of log_{10}(sin(x^2)) is (2xcot(x^2))/(ln(10))
What is the first derivative of log_{10}(sin(x^2)) ?
The first derivative of log_{10}(sin(x^2)) is (2xcot(x^2))/(ln(10))