Derivatives

Finding the nth Derivative

Find the second derivative of the function. 6)

s = \frac{t^{8} + 9 t + 8}{t^{2}}

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Finding the Derivative Using Product Rule

Calculate

\frac{d y}{d x}

. You need not expand your answer.

\frac{y = (3 x^{2} + x) (x - x^{2})}{d x} = ◻

Submit Answer

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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^{'} (0)

. Write your answer as an integer or

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Finding the Derivative Using Chain Rule

Calculate the derivative of the following function.

y = \tan (e^{x})

\frac{d y}{d x} =

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Use Logarithmic Differentiation to Find the Derivative

Let

f (x) = x^{7 x}

. Use logarithmic differentiation to determine the derivative.

f^{'} (x) =

f^{'} (1) =

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Finding the Derivative

Find equations of the tangent line and normal line to the curve

y = 14 \cos x

at the point

(π / 3, 7)

. The derivative

y^{'} (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 =

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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)

{\frac{d y}{d x} |}_{(2, - 3)} =

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Using the Limit Definition to Find the Derivative

a. Use the definition

m_{\tan} = lim_{h \to 0} \frac{f (a + h) - f (a)}{h}

to find the slope of the line tangent to the graph of

f

P

. b. Determine an equation of the tangent line at

P

f (x) = x^{4}, P (2, 16)

m_{\tan} = 32

y =

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Evaluating the Derivative

Determine the rate of change on the interval

(3, 7)

using the graph below:

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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.

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Finding the Linearization

Find the linearization of the function

f (x) = x^{3} + 2 x^{2} - 3 x + 1

x = 2

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Finding the nth Derivative

Finding the Derivative Using Product Rule

Finding the Derivative Using Quotient Rule

Finding the Derivative Using Chain Rule

Use Logarithmic Differentiation to Find the Derivative

Finding the Derivative

Implicit Differentiation

Using the Limit Definition to Find the Derivative

Evaluating the Derivative

Finding Where dy/dx is Equal to Zero

Finding the Linearization

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