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Calculus
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Use the limit definition of the derivative to find \(\displaystyle f'(x)\) if \(\displaystyle f(x)= 7 x^{2} - 4 x - 1\).

Using the difference quotient \(\displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}\) gives \(\displaystyle \lim_{h\to 0}\frac{(- 4 h - 4 x + 7 \left(h + x\right)^{2} - 1)-(7 x^{2} - 4 x - 1)}{h}\) Expanding and collecting like terms gives \(\displaystyle \lim_{h \to 0}\frac{7 h^{2} + 14 h x - 4 h}{h}\) Factoring out \(\displaystyle h\) and reducing gives \(\displaystyle \lim_{h\to 0}7 h + 14 x - 4\). Evaluating the limit gives \(\displaystyle f'(x)=14 x - 4\)

Download \(\LaTeX\) 

\begin{question}Use the limit definition of the derivative to find $f'(x)$ if $f(x)= 7 x^{2} - 4 x - 1$. 
    \soln{9cm}{Using the difference quotient $\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$ gives $\lim_{h\to 0}\frac{(- 4 h - 4 x + 7 \left(h + x\right)^{2} - 1)-(7 x^{2} - 4 x - 1)}{h}$ Expanding and collecting like terms gives $\lim_{h \to 0}\frac{7 h^{2} + 14 h x - 4 h}{h}$ Factoring out $h$ and reducing gives $\lim_{h\to 0}7 h + 14 x - 4$. Evaluating the limit gives $f'(x)=14 x - 4$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Use the limit definition of the derivative to find  <img class="equation_image" title=" \displaystyle f'(x) " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20" alt="LaTeX:  \displaystyle f'(x) " data-equation-content=" \displaystyle f'(x) " />  if  <img class="equation_image" title=" \displaystyle f(x)= 7 x^{2} - 4 x - 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D%207%20x%5E%7B2%7D%20-%204%20x%20-%201%20" alt="LaTeX:  \displaystyle f(x)= 7 x^{2} - 4 x - 1 " data-equation-content=" \displaystyle f(x)= 7 x^{2} - 4 x - 1 " /> . </p> </p>
HTML for Canvas
<p> <p>Using the difference quotient  <img class="equation_image" title=" \displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bh%20%5Cto%200%7D%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} " data-equation-content=" \displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} " />  gives  <img class="equation_image" title=" \displaystyle \lim_{h\to 0}\frac{(- 4 h - 4 x + 7 \left(h + x\right)^{2} - 1)-(7 x^{2} - 4 x - 1)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bh%5Cto%200%7D%5Cfrac%7B%28-%204%20h%20-%204%20x%20%2B%207%20%5Cleft%28h%20%2B%20x%5Cright%29%5E%7B2%7D%20-%201%29-%287%20x%5E%7B2%7D%20-%204%20x%20-%201%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \lim_{h\to 0}\frac{(- 4 h - 4 x + 7 \left(h + x\right)^{2} - 1)-(7 x^{2} - 4 x - 1)}{h} " data-equation-content=" \displaystyle \lim_{h\to 0}\frac{(- 4 h - 4 x + 7 \left(h + x\right)^{2} - 1)-(7 x^{2} - 4 x - 1)}{h} " />  Expanding and collecting like terms gives  <img class="equation_image" title=" \displaystyle \lim_{h \to 0}\frac{7 h^{2} + 14 h x - 4 h}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bh%20%5Cto%200%7D%5Cfrac%7B7%20h%5E%7B2%7D%20%2B%2014%20h%20x%20-%204%20h%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \lim_{h \to 0}\frac{7 h^{2} + 14 h x - 4 h}{h} " data-equation-content=" \displaystyle \lim_{h \to 0}\frac{7 h^{2} + 14 h x - 4 h}{h} " />  Factoring out  <img class="equation_image" title=" \displaystyle h " src="/equation_images/%20%5Cdisplaystyle%20h%20" alt="LaTeX:  \displaystyle h " data-equation-content=" \displaystyle h " />  and reducing gives  <img class="equation_image" title=" \displaystyle \lim_{h\to 0}7 h + 14 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bh%5Cto%200%7D7%20h%20%2B%2014%20x%20-%204%20" alt="LaTeX:  \displaystyle \lim_{h\to 0}7 h + 14 x - 4 " data-equation-content=" \displaystyle \lim_{h\to 0}7 h + 14 x - 4 " /> . Evaluating the limit gives  <img class="equation_image" title=" \displaystyle f'(x)=14 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%3D14%20x%20-%204%20" alt="LaTeX:  \displaystyle f'(x)=14 x - 4 " data-equation-content=" \displaystyle f'(x)=14 x - 4 " /> </p> </p>