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Popular Calculus >

derivative of ln(x^2sin(x))

Solution

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

Solution

\frac{2 sin ( x )}{x} + 2 cos (x) ln (x)

Solution steps

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = ln (x^{2}), g = sin (x)

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

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

\frac{d}{d x} (sin (x)) = cos (x)

= \frac{2}{x} sin (x) + cos (x) ln (x^{2})

Simplify

\frac{2}{x} sin (x) + cos (x) ln (x^{2}) : \frac{2 sin ( x )}{x} + 2 cos (x) ln (x)

= \frac{2 sin ( x )}{x} + 2 cos (x) ln (x)

Graph

Popular Examples

integral of ((y^2+1))/(y^2+4)

\int \frac{( y ^{2} + 1 )}{y ^{2} + 4} d y

integral of (xsin(x))/((cos(x))^2)

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

derivative of x^2y^4

\frac{d}{d x} (x^{2} y^{4})

limit as x approaches 2 of x^2-5

x \to 2 lim (x^{2} - 5)

limit as x approaches 0 of (1-e^{-x})/x

x \to 0 lim (\frac{1 - e ^{- x}}{x})

Frequently Asked Questions (FAQ)

What is the derivative of ln(x^2sin(x)) ?
The derivative of ln(x^2sin(x)) is (2sin(x))/x+2cos(x)ln(x)
What is the first derivative of ln(x^2sin(x)) ?
The first derivative of ln(x^2sin(x)) is (2sin(x))/x+2cos(x)ln(x)