Wize University Calculus 1 Textbook > Derivatives

The Quotient Rule

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The Quotient Rule

Our next differentiation shortcut lets us take derivatives of the quotient (division) of two functions.

Quotient Rule

The derivative of a quotient  f(x)g(x)\displaystyle\ \frac{f(x)}{g(x)}, where f(x)f(x) and g(x)g(x) are differentiable functions and g(x)≠0g(x)\ne 0, is:

[f(x)g(x)]′=f′(x)g(x)−f(x)g′(x)[g(x)]2\boxed{\quad \left[\frac{f\left(x\right)}{g\left(x\right)}\right]'=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{\left[g\left(x\right)\right]^2}\quad }




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Example: The Quotient Rule

Find the following derivative

ddx[3xx−2]\displaystyle \frac{d}{dx}\left[{\frac{3x}{x-2}}\right]


ddx[3xx−2]\displaystyle \frac{d}{dx}\left[\frac{3x}{x-2}\right]

=(x−2)×ddx[3x]−(3x)×ddx[x−2](x−2)2\displaystyle =\frac{\left(x-2\right)\times\frac{d}{dx}\left[3x\right]-\left(3x\right)\times\frac{d}{dx}\left[x-2\right]}{\left(x-2\right)^2}

=(x−2)(3)−(3x)(1)(x−2)2\displaystyle =\frac{\left(x-2\right)\left(3\right)-\left(3x\right)\left(1\right)}{\left(x-2\right)^2}

=3x−6−3x(x−2)2\displaystyle =\frac{3x-6-3x}{\left(x-2\right)^2}

=−6(x−2)2\displaystyle =-\frac{6}{\left(x-2\right)^2}

Suppose g(x)\displaystyle g\left(x\right) is differentiable, f(x)=1+x g(x)x2f\left(x\right)=\displaystyle \frac{1+x\ g\left(x\right)}{x^2}, f′(1)=2f'\left(1\right)=2, g(1)=1g\left(1\right)=1, what is g′(1)g'\left(1\right)?

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Extra Practice

The quotient rule

Find the derivative of the following function:

g(x)=x+1x+1\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}

Find the derivative of the following function:

g(x)=x+1x+1\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}

The Quotient Rule

Suppose g(x)g\left(x\right) is differentiable, f(x)=1+x g(x)x2f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}, f′(1)=2f'\left(1\right)=2, g(1)=1g\left(1\right)=1, what is g′(1)g'\left(1\right)?

Compute the derivative of f(x)=x2+3x4/3+1x2+1f(x) =\dfrac{x^2 + 3x^{4/3} + 1}{x^2 + 1}.

Find the derivative with respect to xx of the function x20303+x2030\displaystyle \frac{x^{2030}}{3+x^{2030}}.

The Quotient Rule

Is u′(t)=2(t−1)3(−3t+23)(3t+7)7u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7} the derivative of u(t)=(t−1)4(3t+7)6u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6} ?

The Quotient Rule

If h(x)=f(x)−1x+2h\left(x\right)=\frac{f\left(x\right)-1}{x+2}, f(3)=2 and f′(3)=−6f\left(3\right)=2\ \text{and}\ f'\left(3\right)=-6, calculate the value of h′(3)h'\left(3\right)

Practice: Quotient Rule

Find the derivative of f(x)=x2+3x4/3+1x2+1\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}

Quotient and Product

Find the derivative of g(t)=(1+2t7t)(t2−1)\displaystyle g(t)=\left(\frac{1+2t}{7t}\right)(t^{2}-1)

The quotient rule

Is u′(t)=2(t−1)3(−3t+23)(3t+7)7\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7} the derivative of u(t)=(t−1)4(3t+7)6\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6} ?

Derivatives: Exponential and Logarithmic Functions

Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.

The quotient rule

Given the function f(x)=g(x)h(x)−excos⁡xf\left(x\right)=\dfrac{g\left(x\right)}{h\left(x\right)}-e^x\cos x, and given g(π2)=2g\left(\dfrac{\pi}{2}\right)=2, g′(π2)=3g'\left(\dfrac{\pi}{2}\right)=3, h(π2)=1h\left(\dfrac{\pi}{2}\right)=1, and h′(π2)=2h'\left(\dfrac{\pi}{2}\right)=2, find f′(π2)f'\left(\dfrac{\pi}{2}\right).

Practice: Product Rule*

Suppose g(x)g\left(x\right) is differentiable, f(x)=1+x g(x)x2f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}, f′(1)=2f'\left(1\right)=2, g(1)=1g\left(1\right)=1, what is g′(1)g'\left(1\right)?

The Quotient Rule

Compute the derivative of

f(x)=x2+3x4/3+1x2+1f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}

Derivatives

Find the derivative of the following functions at the given values

a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0

b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0

The Quotient Rule

Find the derivative of g(x)=x2−52x+1g(x) = \frac{x^2 - 5}{2x + 1}

The Quotient and Chain Rules

Let f(x)=x2−6x−3\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}.

Find the derivative of the following function:

g(x)=x+1x+1\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}

Practice: Quotient Rule

Q.\textbf{Q.} Find the derivative of f(x)=x2+3x4/3+1x2+1\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}

Quotient and Product

Find the derivative of g(t)=(1+2t7t)(t2−1)\displaystyle g(t)=\left(\frac{1+2t}{7t}\right)(t^{2}-1)

Derivatives: Exponential and Logarithmic Functions

Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.

The quotient rule

Given the function f(x)=g(x)h(x)−excos⁡xf\left(x\right)=\dfrac{g\left(x\right)}{h\left(x\right)}-e^x\cos x, and given g(π2)=2g\left(\dfrac{\pi}{2}\right)=2, g′(π2)=3g'\left(\dfrac{\pi}{2}\right)=3, h(π2)=1h\left(\dfrac{\pi}{2}\right)=1, and h′(π2)=2h'\left(\dfrac{\pi}{2}\right)=2, find f′(π2)f'\left(\dfrac{\pi}{2}\right).

The quotient rule

Is u′(t)=2(t−1)3(−3t+23)(3t+7)7\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7} the derivative of u(t)=(t−1)4(3t+7)6\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6} ?

The quotient rule

Find the derivative of the following function:

g(x)=x+1x+1\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}

The quotient rule

Is u′(t)=2(t−1)3(−3t+23)(3t+7)7\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7} the derivative of u(t)=(t−1)4(3t+7)6\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6} ?

Derivatives

Find the derivative of the following functions at the given values

a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0

b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0