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Find the derivative of the function
f(x)=(2x-3)/(x^(3)+3x) using the limit definition.

$13$ ) Find the derivative of the function $f(x)=\frac{2 x-3}{x^{3}+3 x}$ using the limit definition.

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Find derivatives of logarithmic functions

Full solution

Q.

13

) Find the derivative of the function

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

using the limit definition.

Define function and limit definition: Question Prompt: Find the derivative of the function $f(x) = \frac{2x-3}{x^3+3x}$ using the limit definition.
Substitute $f(x+h)$ : Step $1$ : Define the function and apply the limit definition of the derivative. $\newline$ $f(x) = \frac{2x-3}{x^3+3x}$ $\newline$ $f'(x) = \lim_{h\to0} \left[\frac{f(x+h) - f(x)}{h}\right]$
Calculate $f(x+h) - f(x)$ : Step $2$ : Substitute $f(x+h)$ in the formula. $\newline$ $f(x+h) = \frac{2(x+h)-3}{(x+h)^3+3(x+h)}$ $\newline$ $= \frac{2x + 2h - 3}{(x^3 + 3x^2h + 3xh^2 + h^3) + 3x + 3h}$
Simplify expression by finding common denominator: Step $3$ : Calculate $f(x+h) - f(x)$ . $f(x+h) - f(x) = \left[\frac{2x + 2h - 3}{(x^3 + 3x^2h + 3xh^2 + h^3) + 3x + 3h}\right] - \left[\frac{2x-3}{x^3+3x}\right]$
Simplify expression by finding common denominator: Step $3$ : Calculate $f(x+h) - f(x)$ . $f(x+h) - f(x) = \frac{2x + 2h - 3}{(x^3 + 3x^2h + 3xh^2 + h^3) + 3x + 3h} - \frac{2x-3}{x^3+3x}$ Step $4$ : Simplify the expression by finding a common denominator. Common denominator = $(x^3 + 3x^2h + 3xh^2 + h^3 + 3x + 3h)(x^3 + 3x) = x^6 + 3x^4 + 3x^3h + x^3h^3 + 9x^2h + 9xh^2 + 3h^4 + 9x + 9h$

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