a) Use the Quotient Rule to find the derivative of the given function. b) Find the derivative by...

Question:

a) Use the Quotient Rule to find the derivative of the given function.

b) Find the derivative by dividing the expressions first.

{eq}y = \frac{x^8}{x^6} \ \text{for} \ x \neq 0 {/eq}

Differentiation Rules


Differentiating complex functions, we may need to use the quotient rule

{eq}\displaystyle \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)= \frac{g(x)\dfrac{d}{dx}(f(x))-f(x)\dfrac{d}{dx}(g(x))}{g^2(x)}, {/eq} or the product rule, if we have a product of functions,

{eq}\displaystyle \frac{d}{dx}\left[f(x)g(x)\right]= g(x)\frac{d}{dx}(f(x))+ f(x)\frac{d}{dx}(g(x)). {/eq}

Polynomial functions are differentiated based on the following power rule {eq}\displaystyle \frac{d}{dx}(x^n)=nx^{n-1}, n\in\mathbf{R}, {/eq}

Answer and Explanation:


a) Using the Quotient Rule to differentiate the function {eq}\displaystyle y = \frac{x^8}{x^6} \ \text{for} \ x \neq 0 {/eq}

we will obtain the derivative as follows.

{eq}\displaystyle \begin{align}y' &= \frac{d}{dx}\left(\frac{x^8}{x^6}\right) \ \text{for} \ x \neq 0\\ &=\frac{x^6\dfrac{d}{dx}(x^8)-x^8\dfrac{d}{dx}(x^6)}{(x^6)^2}\\ &=\frac{8x^6\cdot x^7-6x^8\cdot x^5}{x^{12}}\\ &=\frac{8x^{13}-6x^{13}}{x^{12}}\\ &=\frac{2x^{13}}{x^{12}}=\boxed{2x}, \text{ because }x\neq 0. \end{align} {/eq}


b) The function {eq}\displaystyle y = \frac{x^8}{x^6} \ \text{for} \ x \neq 0 {/eq} can be simplified, first, because {eq}\displaystyle x \neq 0 {/eq} and then differentiate, as below.

{eq}\displaystyle \begin{align}y &= \frac{x^8}{x^6} =x^2\ \text{for} \ x \neq 0\\ \implies y'(x)&=\frac{d}{dx}(x^2)=\boxed{2x}. \end{align} {/eq}


Learn more about this topic:

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When to Use the Quotient Rule for Differentiation

from Math 104: Calculus

Chapter 8 / Lesson 8
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