Find the derivatives of the following functions: a) f(x) = x^{3} - 4x + \sqrt{x} - 9 b) f(x) =...

Question:

Find the derivatives of the following functions:

{eq}a)\ f(x)=x^{3}-4x+\sqrt{x}-9 {/eq}

{eq}b)\ f(x)=\ln(3x^{2}-x) {/eq}

{eq}c)\ f(x)=xe^{2x} {/eq}

{eq}d)\ f(x)=(x+x^{2})^{\frac{3}{2}} {/eq}

{eq}e)\ f(x)=\frac{x}{1-x} {/eq}

Differentiation:

Differentiation is the way of finding the derivative or rate of change of the dependent variable with respect to the independent variable.

Let {eq}y = f(x){/eq}, then differentiating both sides with respect to {eq}x{/eq}, we get

{eq}\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}f(x) = f'(x). {/eq}

We apply chain rule in given problem:

{eq}\frac{{df(t)}}{{dx}} = \frac{{df}}{{dt}}\cdot \frac{{dt}}{{dx}}. {/eq}

We recall the following formulas of given differentiation:

{eq}\eqalign{ \frac{d}{{dx}}\left( x^n \right) &= nx^{n-1}, \cr \frac{d}{{dx}}\left( {e^x} \right) &= e^x \cr} {/eq}

{eq}\displaystyle \frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}. {/eq}

We also use the multiplication and division rule of differentiation:

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

{eq}\frac{d}{{dx}}\left( {\frac{{f(x)}}{{g(x)}}} \right) = \frac{{g(x) \cdot \frac{d}{{dx}}\left( {f(x)} \right) - f(x) \cdot \frac{d}{{dx}}\left( {g(x)} \right)}}{{{{\left( {g(x)} \right)}^2}}}. {/eq}

Answer and Explanation: 1

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Solution (a):

Given, {eq}f(x)=x^{3}-4x+\sqrt{x}-9, {/eq}

differentiating both sides with respect to {eq}x,{/eq}

{eq}\displaystyle \eqalign{ &...

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Differentiation Strategy: Definition & Examples

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Chapter 7 / Lesson 15
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What is a differentiation strategy? Learn about focused differentiation strategy, broad differentiation strategy, and other differentiation strategy examples.


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