What is the derivative of log_{4}((x^2+2x^3)) ?

The derivative of log_{4}((x^2+2x^3)) is (3(x+1))/(ln(2)x(x+2))

What is the first derivative of log_{4}((x^2+2x^3)) ?

The first derivative of log_{4}((x^2+2x^3)) is (3(x+1))/(ln(2)x(x+2))

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\frac{d}{d x} (lo g_{4} ((x^{2} + 2 x)^{3}))

\frac{3 ( x + 1 )}{ln ( 2 ) x ( x + 2 )}

Solution steps

\frac{d}{d x} (lo g_{4} ((x^{2} + 2 x)^{3}))

Apply log rule:

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

= \frac{d}{d x} \frac{ln ( ( x ^{2} + 2 x ) ^{3} )}{ln ( 4 )}

Take the constant out:

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

= \frac{1}{ln ( 4 )} \frac{d}{d x} (ln ((x^{2} + 2 x)^{3}))

Apply the chain rule:

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

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

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

= \frac{1}{ln ( 4 )} \cdot \frac{1}{( x ^{2} + 2 x ) ^{3}} \cdot 3 (x^{2} + 2 x)^{2} (2 x + 2)

Simplify

\frac{1}{ln ( 4 )} \cdot \frac{1}{( x ^{2} + 2 x ) ^{3}} \cdot 3 (x^{2} + 2 x)^{2} (2 x + 2) : \frac{3 ( x + 1 )}{ln ( 2 ) x ( x + 2 )}

= \frac{3 ( x + 1 )}{ln ( 2 ) x ( x + 2 )}

f (x) = x 8 - x^{2}

\int (1 + cos (y)) d y

\frac{π}{6} cos (x)

2 - x^{6}

\int 5 x - x^{2} - x d x

What is the derivative of log_{4}((x^2+2x^3)) ?
The derivative of log_{4}((x^2+2x^3)) is (3(x+1))/(ln(2)x(x+2))
What is the first derivative of log_{4}((x^2+2x^3)) ?
The first derivative of log_{4}((x^2+2x^3)) is (3(x+1))/(ln(2)x(x+2))