# (1) Find the derivative of f(x)=\frac{5}{3x+5} using the limit definition of derivative. (2) Find...

## Question:

(1) Find the derivative of {eq}f(x)=\frac{5}{3x+5} {/eq} using the limit definition of derivative. (2) Find the derivative of {eq}f(x)=\sqrt{3x+5} {/eq} using the limit definition of derivative.

## Differentiation by the Limit Derivative Method:

{eq}\displaystyle \text{We can find any derivative by using the limit definition of a derivative.}\\[10pt] \text{This is the process of finding the derivative of function f(x) by using the following definition}\\[10pt] f'(x)=\lim_{h\to0} \frac{f(x+h)-f(x)}{h}\\ {/eq}

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## Answer and Explanation:

we can apply the limit definition of a derivative to differentiate this function:

{eq}\displaystyle{\text{Let}f(x)=\frac{5}{3x+5}\\[10pt] \text{Then...

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View this answerwe can apply the limit definition of a derivative to differentiate this function:

{eq}\displaystyle{\text{Let}f(x)=\frac{5}{3x+5}\\[10pt] \text{Then, } f(x+h)=\frac{5}{3(x+h)+5}\\[10pt] \text{Now, }\frac{d}{dx}(f(x))=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\\[10pt] \frac{d}{dx}(f(x))=\lim_{h \to 0}\frac{\frac{5}{3(x+h)+5}- \frac{5}{3x+5}}{h}\\[10pt] \frac{d}{dx}(f(x))=\lim_{h \to 0}\frac{5(3x+5)-5(3x+3h+5)}{h(3x+3h+5)(3x+5)}\\[10pt] \frac{d}{dx}(f(x)) =\lim_{h \to 0}\frac{15x+25-15x-15h-25}{h(3x+3h+5)(3x+5)}\hspace{30pt}\text{By simplifying the numerator}\\[10pt] \frac{d}{dx}(f(x)) =\lim_{h \to 0}\frac{-15h}{h(3x+3h+5)(3x+5)}\\[10pt] \frac{d}{dx}(f(x)) =\lim_{h \to 0}\frac{-15}{(3x+3h+5)(3x+5)}\\[10pt] \text{Now, by applying }\lim_{h \to 0}, \text{ we get}\\[10pt] \frac{d}{dx}(f(x)) = \frac{-15}{(3x+5)(3x+5)}\\[10pt] \text{Therefore, } \boxed{\frac{d}{dx}(f(x)) = \frac{-15}{(3x+5)^2}}\\} {/eq}

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