Quotient Rule Practice Problems

Problem: Quotient Rule #1

Find the derivative of:

$\Large{f(x) = \frac{4x^2}{2x-3}}$

Solution

Step 1:

We notice that the given function is a quotient of two other functions. To find the derivative of this, we can use the quotient rule. First, let’s label the two functions as:

$\large{h(x) = 4x^2}$ and $\large{g(x) = 2x-3}$

To use the quotient rule, we also need the derivative of these functions. We can easily find the derivatives as:

$\large{h'(x) = 8x}$ and $\large{g'(x) = 2}$

Step 2:

The quotient rule formula states that:

\large{\frac{d}{dx}\frac{f(x)}{g(x)} = \frac{g(x)f'(x) – f(x)g'(x)}{g(x)^2}}

Thus, substituting into the quotient rule formula, we have:

\large{\frac{d}{dx}(\frac{4x^2}{2x-3}) = \frac{(2x-3)(8x) \:- \:(4x^2)(2)}{(2x-3)^2}}

Step 3:

Simplifying, we have:

\Large{ = \frac{16x^2-24x-8x^2}{(2x-3)^2}}

\Large{ = \frac{8x^2-24x}{(2x-3)^2}}

Thus, this is the final answer for the derivative. So:

\Large{f'(x) = \frac{8x^2-24x}{(2x-3)^2}}

Solution Complete!

Problem: Quotient Rule #2

Find the derivative of:

$\Large{y = \frac{x^2-3x+1}{2x^2+5}}$

Problem: Quotient Rule #3

Find the derivative of:

$h(x) =\Large{ \frac{e^{x}-x}{\sin(x)}}$

Problem: Quotient Rule #4

Find the derivative of:

$y = \Large{\frac{\sqrt{x}}{2x+1}}$

Problem: Quotient Rule #5

Find the derivative of:

$\Large{f(x) = \frac{\ln(x)-1}{\cot(x)}}$

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