Quotient Rule Calculator

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Example

Created on 2024-06-20Asked by Alexander Carter (Solvelet student)

Find the derivative of

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

using the quotient rule.

Solution

To find the derivative of

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

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

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

2. Identify

u

and

v

u = x^2, \quad v = e^x.

3. Compute the derivatives of

u

and

v

u' = 2x, \quad v' = e^x.

4. Apply the quotient rule:

\begin{aligned} f'(x) &= \frac{{2xe^x - x^2e^x}}{{(e^x)^2}} \\ &= \frac{{2xe^x - x^2e^x}}{{e^{2x}}}. \end{aligned}

Therefore, the derivative of

f(x)

f'(x) = \frac{{2xe^x - x^2e^x}}{{e^{2x}}}

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Liam Hill on Solvelet

1. Find the derivative of the function

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

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

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

at the point

(2, -5)

DefinitionQuotient rule is used for calculating the derivative of the quotient of two functions. If u(x) and v(x) are differentiable functions, then if vu is differentiable with the derivative(vu)′=v2u′−u(v′vu)=(v2u′−uv′). E.g., for f(x)=sin(x)x2, the derivative would be f′(x)=sin2(x)2xsin(x)−x2cos(x).

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