AMAT $112$ Calculus $1$

Written Assignment $2$

Question $1)$ Find the first derivatives of $f(x)$ using the definition of the derivative $($Note: You may decide which variation of the difference quotient to use$)(4$ points each$)$

Recall the definition of the derivative formulas:

$f^{\text{'}}(a)=\underset{xa}{lim}\frac{f(x)-f(a)}{x-a}$ and $f^{\text{'}}(x)=\underset{h0}{lim}\frac{f(xh)-f(x)}{h}$

a$)f(x)=\sqrt[\phantom{2}]{x3}$

b$)f(x)=2x-6x^2$ at $x=1$

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AMAT $112$ Calculus $1$

Written Assignment $2$

Question $2)$ Sketch a graph of the first derivative of the function given below on the same set of axes. What do increasing$/$decreasing$/$maximums$/$minimums look like in terms of your derivative function? Be specific. $(7$ points$)$

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Question $3)$ Compute the first derivatives of the following using the rule specified: $(3$ points each $a-d,$ e $5$ points$).$ Do not simplify your answer.

a$)($Power Rule$)f(x)=12x^3-4x^{7}3x^8-\frac{1}{x^2}$

b$)($Product Rule$)g(x)=e^{x}sin(x)$

c$)($Quotient Rule$)h(x)=\frac{5x^6}{cos(x)}$

d$)($Chain Rule$)k(x)=4(9x^3-2x^{5}6{)}^5$

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is form.

e$)$ Any rules necessary$)m(x)=ln(\frac{sin(6x^3)}{2x{e}^{4x}})$

Question $4)$ Find the equation of the tangent line of $2x^3{y}^2=x^{2}y$ at the point $(-1,-1)($Hint: Use implicit differentiation to find the derivative$)(4$ points$)$

Question $5)$ Find $f^{\text{'}}(x)$ if $f(x)=co{s}^{-1}(8x^3).$ Does the derivative exist at $(\frac{1}{2},0)?$ Justify. $($Note: Do not just look up the derivative of the inverse cosine function; explicitly solve for your answer$)(4$ points$)$