Quotient rule of differentiation Calculator

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Example

Created on 2024-06-20Asked by Victoria Hill (Solvelet student)

Find the derivative of

f(x) = \frac{{x^2}}{{\sin(x)}}

using the quotient rule of differentiation.

Solution

To find the derivative of

f(x) = \frac{{x^2}}{{\sin(x)}}

using the quotient rule of differentiation: 1. Apply the quotient rule:

\left( \frac{{u}}{{v}} \right)' = \frac{{u'v - uv'}}{{v^2}}.

2. Identify

u

and

v

u = x^2, \quad v = \sin(x).

3. Compute the derivatives of

u

and

v

u' = 2x, \quad v' = \cos(x).

4. Apply the quotient rule:

\begin{aligned} f'(x) &= \frac{{2x \cdot \sin(x) - x^2 \cdot \cos(x)}}{{(\sin(x))^2}} \\ &= \frac{{2x \sin(x) - x^2 \cos(x)}}{{\sin^2(x)}}. \end{aligned}

Therefore, the derivative of

f(x)

f'(x) = \frac{{2x \sin(x) - x^2 \cos(x)}}{{\sin^2(x)}}

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Olivia Jackson on Solvelet

1. Find the derivative of the function

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

.2. Calculate the slope of the tangent line to the curve

y = \frac{x - 2}{x + 3}

at the point

(1, -4)

DefinitionA formula that used to evaluate the derivative of a function which is a division of two functions is called the quotient rule of differentiation. Which is read as (g(x)f(x) )'=g(x)2f'(x)g(x)−f(x)g'(x) . For instance, h(x)=xex which has a derivative equal to h′(x)=x2ex(x−1).

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