Finding the nth Derivative
Find the second derivative of the function.
6) $s=\frac{t^{8}+9 t+8}{t^{2}}$
Finding the Derivative Using Product Rule
Calculate $\frac{d y}{d x}$. You need not expand your answer.
\[
\frac{y=\left(3 x^{2}+x\right)\left(x-x^{2}\right)}{d x}=\square
\]
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Finding the Derivative Using Quotient Rule
Let $p$ and $q$ be piecewise linear functions given by their respective graphs below.
Let $r(x)=\frac{q(x)}{p(x)}$. Determine $r^{\prime}(0)$. Write your answer as an integer or
Finding the Derivative Using Chain Rule
Calculate the derivative of the following function.
\[
y=\tan \left(e^{x}\right)
\]
\[
\frac{d y}{d x}=
\]
Use Logarithmic Differentiation to Find the Derivative
Let $f(x)=x^{7 x}$. Use logarithmic differentiation to determine the derivative.
\[
f^{\prime}(x)=
\]
\[
f^{\prime}(1)=
\]
Finding the Derivative
Find equations of the tangent line and normal line to the curve $y=14 \cos x$ at the point $(\pi / 3,7)$.
The derivative $y^{\prime}(x)=$
The slope of the tangent line is $m_{1}=$
The equation of the tangent line is $y=$
The slope of the normal line is $m_{2}=$
The equation of the normal line is $y=$
Implicit Differentiation
Evaluate the derivative of the following function at the given point.
\[
18 x^{3} y^{2}-6 y^{3}=1,458 ;(2,-3)
\]
\[
\left.\frac{d y}{d x}\right|_{(2,-3)}=
\]
Using the Limit Definition to Find the Derivative
a. Use the definition $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ to find the slope of the line tangent to the graph of $f$ at $P$.
b. Determine an equation of the tangent line at $P$.
\[
f(x)=x^{4}, P(2,16)
\]
a. $m_{\tan }=32$
b. $y=$
Evaluating the Derivative
Determine the rate of change on the interval $(3,7)$ using the graph below:
Finding Where dy/dx is Equal to Zero
Use derivatives to find the critical points and inflection points of
\[
f(x)=x^{5}-10 x^{3}-12
\]
Find all critical and inflection points.
Finding the Linearization
Find the linearization of the function \(f(x) = x^3 + 2x^2 - 3x + 1\) at \(x = 2\).