Higher-order derivatives Calculator

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Example

Created on 2024-06-20Asked by Elijah Nelson (Solvelet student)

Find the fourth derivative of

f(x) = \ln(x^2 + 1)

Solution

To find the fourth derivative of

f(x) = \ln(x^2 + 1)

: First, calculate the first derivative:

f'(x) = \frac{d}{dx} \left( \ln(x^2 + 1) \right) = \frac{2x}{x^2 + 1}

Next, calculate the second derivative:

f''(x) = \frac{d}{dx} \left( \frac{2x}{x^2 + 1} \right) = \frac{(2)(x^2 + 1) - (2x)(2x)}{(x^2 + 1)^2} = \frac{2 - 2x^2}{(x^2 + 1)^2} = \frac{2(1 - x^2)}{(x^2 + 1)^2}

Next, calculate the third derivative:

f'''(x) = \frac{d}{dx} \left( \frac{2(1 - x^2)}{(x^2 + 1)^2} \right)

Using the quotient rule:

f'''(x) = \frac{2(-2x)(x^2 + 1)^2 - 2(1 - x^2)2(x^2 + 1)2x}{(x^2 + 1)^4}

= \frac{-4x(x^2 + 1)^2 - 8x(1 - x^2)(x^2 + 1)}{(x^2 + 1)^4}

= \frac{-4x(x^2 + 1) - 8x(1 - x^2)}{(x^2 + 1)^3}

= \frac{-4x^3 - 4x - 8x + 8x^3}{(x^2 + 1)^3}

= \frac{4x^3 - 12x}{(x^2 + 1)^3}

Finally, calculate the fourth derivative:

f''''(x) = \frac{d}{dx} \left( \frac{4x^3 - 12x}{(x^2 + 1)^3} \right)

Using the quotient rule again:

f''''(x) = \frac{(12x^2 - 12)(x^2 + 1)^3 - (4x^3 - 12x)3(x^2 + 1)^2(2x)}{(x^2 + 1)^6}

= \frac{12x^2 - 12 - 24x^3 + 72x^2}{(x^2 + 1)^4}

= \frac{12x^2 - 12 + 24x^3 - 72x^2}{(x^2 + 1)^4}

= \frac{-48x^2 - 12 + 24x^3}{(x^2 + 1)^4}

= \frac{24x^3 - 60x^2 - 12}{(x^2 + 1)^4}

Therefore, the fourth derivative of

f(x) = \ln(x^2 + 1)

is:

f''''(x) = \frac{24x^3 - 60x^2 - 12}{(x^2 + 1)^4}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mia Taylor on Solvelet

1. Compute the second derivative of the function

f(x) = x^3

with respect to

x

,2. Show that the function

g(x) = \sin(x)

is a homeomorphism between the interval

[-\frac{\pi}{2}, \frac{\pi}{2}]

and the open interval

(-1, 1)

DefinitionA higher-order differential equation is a differential equation that contains derivatives of an unknown function of order higher than one. Equations which describe how dynamic systems and phenomena develop over time, and are used in physics, engineering, and applied mathematics. Any higher-order differential equation has to slug in an initial or boundary condition for each derivative of various order. But, this second-order differential equation dx2d2y+4dxdy+4y=0

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