12.4 Derivatives  (Page 3/18)

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Notations for the derivative

The equation of the derivative of a function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is written as $\text{\hspace{0.17em}}{y}^{\prime }={f}^{\prime }\left(x\right),$ where $\text{\hspace{0.17em}}y=f\left(x\right).\text{\hspace{0.17em}}$ The notation $\text{\hspace{0.17em}}{f}^{\prime }\left(x\right)\text{\hspace{0.17em}}$ is read as “ ” Alternate notations for the derivative include the following:

${f}^{\prime }\left(x\right)={y}^{\prime }=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}f\left(x\right)=Df\left(x\right)$

The expression $\text{\hspace{0.17em}}{f}^{\prime }\left(x\right)\text{\hspace{0.17em}}$ is now a function of $\text{\hspace{0.17em}}x$ ; this function gives the slope of the curve $\text{\hspace{0.17em}}y=f\left(x\right)\text{\hspace{0.17em}}$ at any value of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ The derivative of a function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ at a point $\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ is denoted $\text{\hspace{0.17em}}{f}^{\prime }\left(a\right).$

Given a function $\text{\hspace{0.17em}}f,$ find the derivative by applying the definition of the derivative.

1. Calculate $\text{\hspace{0.17em}}f\left(a+h\right).$
2. Calculate $\text{\hspace{0.17em}}f\left(a\right).$
3. Substitute and simplify $\text{\hspace{0.17em}}\frac{f\left(a+h\right)-f\left(a\right)}{h}.$
4. Evaluate the limit if it exists: $\text{\hspace{0.17em}}{f}^{\prime }\left(a\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(a+h\right)-f\left(a\right)}{h}.$

Finding the derivative of a polynomial function

Find the derivative of the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}-3x+5\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=a.$

We have:

Substitute $\text{\hspace{0.17em}}f\left(a+h\right)={\left(a+h\right)}^{2}-3\left(a+h\right)+5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(a\right)={a}^{2}-3a+5.$

Find the derivative of the function $\text{\hspace{0.17em}}f\left(x\right)=3{x}^{2}+7x\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=a.$

${f}^{\prime }\left(a\right)=6a+7$

Finding derivatives of rational functions

To find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned.

Finding the derivative of a rational function

Find the derivative of the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{3+x}{2-x}\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=a.$

Find the derivative of the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{10x+11}{5x+4}\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=a.$

${f}^{\prime }\left(a\right)=\frac{-15}{{\left(5a+4\right)}^{2}}$

Finding derivatives of functions with roots

To find derivatives of functions with roots, we use the methods we have learned to find limits of functions with roots, including multiplying by a conjugate.

Finding the derivative of a function with a root

Find the derivative    of the function $\text{\hspace{0.17em}}f\left(x\right)=4\sqrt{x}\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=36.$

We have

Multiply the numerator and denominator by the conjugate: $\text{\hspace{0.17em}}\frac{4\sqrt{a+h}+4\sqrt{a}}{4\sqrt{a+h}+4\sqrt{a}}.$

Find the derivative of the function $\text{\hspace{0.17em}}f\left(x\right)=9\sqrt{x}$ at $\text{\hspace{0.17em}}x=9.$

$\frac{3}{2}$

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