What is the derivative of ln(cos^2(x+5x^3)) ?

The derivative of ln(cos^2(x+5x^3)) is (-sin(2x)+15x^2)/(cos^2(x)+5x^3)

What is the first derivative of ln(cos^2(x+5x^3)) ?

The first derivative of ln(cos^2(x+5x^3)) is (-sin(2x)+15x^2)/(cos^2(x)+5x^3)

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\frac{d}{d x} (ln (cos^{2} (x) + 5 x^{3}))

\frac{- sin ( 2 x ) + 15 x ^{2}}{cos ^{2} ( x ) + 5 x ^{3}}

Solution steps

\frac{d}{d x} (ln (cos^{2} (x) + 5 x^{3}))

Apply the chain rule:

\frac{1}{cos ^{2} ( x ) + 5 x ^{3}} \frac{d}{d x} (cos^{2} (x) + 5 x^{3})

= \frac{1}{cos ^{2} ( x ) + 5 x ^{3}} \frac{d}{d x} (cos^{2} (x) + 5 x^{3})

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

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

Simplify

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

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

y^{^{''}} = - k^{2} y

\frac{d}{d x} (a x^{3} + b x^{2} + c x + d)

f (x) = \frac{2 x ^{3}}{3} - 12 x

\frac{d}{d x} ((5 x^{3} + 4) (3 x^{7} - 5))

2 x^{2} - 3 x, (0, 5)

What is the derivative of ln(cos^2(x+5x^3)) ?
The derivative of ln(cos^2(x+5x^3)) is (-sin(2x)+15x^2)/(cos^2(x)+5x^3)
What is the first derivative of ln(cos^2(x+5x^3)) ?
The first derivative of ln(cos^2(x+5x^3)) is (-sin(2x)+15x^2)/(cos^2(x)+5x^3)