What is the derivative of ln({g}(x(3x^2))) ?

The derivative of ln({g}(x(3x^2))) is (x derivative of {g}(x)+2{g}(x))/(x{g)(x)}

What is the first derivative of ln({g}(x(3x^2))) ?

The first derivative of ln({g}(x(3x^2))) is (x derivative of {g}(x)+2{g}(x))/(x{g)(x)}

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\frac{d}{d x} (ln (g (x) (3 x^{2})))

\frac{x \frac{d}{d x} ( g ( x ) ) + 2 g ( x )}{x g ( x )}

Solution steps

\frac{d}{d x} (ln (g (x) (3 x^{2})))

Apply the chain rule:

\frac{1}{g ( x ) \cdot 3 x ^{2}} \frac{d}{d x} (g (x) \cdot 3 x^{2})

= \frac{1}{g ( x ) \cdot 3 x ^{2}} \frac{d}{d x} (g (x) \cdot 3 x^{2})

\frac{d}{d x} (g (x) \cdot 3 x^{2}) = 3 (\frac{d}{d x} (g (x)) x^{2} + 2 x g (x))

= \frac{1}{g ( x ) \cdot 3 x ^{2}} \cdot 3 (\frac{d}{d x} (g (x)) x^{2} + 2 x g (x))

Simplify

\frac{1}{g ( x ) \cdot 3 x ^{2}} \cdot 3 (\frac{d}{d x} (g (x)) x^{2} + 2 x g (x)) : \frac{x \frac{d}{d x} ( g ( x ) ) + 2 g ( x )}{x g ( x )}

= \frac{x \frac{d}{d x} ( g ( x ) ) + 2 g ( x )}{x g ( x )}

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

\int \frac{y}{( y + 5 ) ( 2 y - 1 )} d y

f (t) = ln (1 + 2 \cdot t)

\int \frac{7}{( x - 10 ) ^{7}} d x

\int \frac{x}{3 x - 2} d x

What is the derivative of ln({g}(x(3x^2))) ?
The derivative of ln({g}(x(3x^2))) is (x derivative of {g}(x)+2{g}(x))/(x{g)(x)}
What is the first derivative of ln({g}(x(3x^2))) ?
The first derivative of ln({g}(x(3x^2))) is (x derivative of {g}(x)+2{g}(x))/(x{g)(x)}