Question 1

Find an equation of the tangent line to the graph of $y = \frac { \ln x } { x }$ at the point $\left( e ^ { 2 } , \frac { 2 } { e ^ { 2 } } \right)$ .

Accepted Answer

The answer of Find an equation of the tangent line...

Question 2

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for the parametric curve given by $x = t ^ { 4 } - t ^ { 2 } + t , y = \sqrt [ 3 ] { t }$

Accepted Answer

The answer of Find \(\frac { d ^ { 2...

Question 3

If $f ( x ) = x ^ { 5 }$ , then, $f ^ { ( 5 ) } ( 3 )$ .

Accepted Answer

A) 90 
B) 360 
C) 30 
D) 60 
E) 20
F)72
G)120
H)240 
A) 90 
B) 360 
C) 30 
D) 60 
E) 20
F)72
G)120
H)240

Question 4

Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 2 x }$ .

Accepted Answer

A) 3 
B) 0 
C)  $\frac { 1 } { 3 }$ 
D) 4 
E)  $\frac { 1 } { 4 }$ 
F)2
G) $\frac { 1 } { 2 }$ 
H)1 
A) 3 
B) 0 
C)  $\frac { 1 } { 3 }$ 
D) 4 
E)  $\frac { 1 } { 4 }$ 
F)2
G) $\frac { 1 } { 2 }$ 
H)1

Question 5

If you were to drag a 200 pound object along a horizontal surface, then the smallest force F that is necessary to drag that object is given by $F ( \theta ) = \frac { 200 k } { \cos \theta + k \sin \theta }$ where $\theta$ is the angle (in radians) your arm makes with the ground and k > 0 is the coefficient of friction for the surface. Find a formula for $F ^ { \prime } ( \theta ) \text { for } 0 \leq \theta \leq \frac { \pi } { 2 }$

Accepted Answer

The answer of If you were to drag a 200...

Question 6

Find $\frac { d y } { d x }$ .(a) $y = \left( \frac { x + 1 } { x - 3 } \right) ^ { 4 }$ (b) $y = e ^ { 2 x } + e ^ { 99 }$ (c) $y = x e ^ { x ^ { 2 } + 7 }$ (d) $y = \tan \left( \sin x ^ { 2 } \right)$

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 7

Find the exact value of $\sin ^ { - 1 } \left( \sin \frac { 5 \pi } { 6 } \right)$ .

Accepted Answer

A)  $- \frac { 5 \pi } { 6 }$ 
B)  $- \frac { 2 \pi } { 3 }$ 
C)  $- \frac { \pi } { 6 }$ 
D) 0 
E)  $\frac { \pi } { 6 }$
F) $\frac { \pi } { 3 }$ 
G) $\frac { 2 \pi } { 3 }$ 
H) $\frac { 5 \pi } { 6 }$ 
A)  $- \frac { 5 \pi } { 6 }$ 
B)  $- \frac { 2 \pi } { 3 }$ 
C)  $- \frac { \pi } { 6 }$ 
D) 0 
E)  $\frac { \pi } { 6 }$
F) $\frac { \pi } { 3 }$ 
G) $\frac { 2 \pi } { 3 }$ 
H) $\frac { 5 \pi } { 6 }$

Question 8

Find the linear approximation to $f ( x ) = e ^ { - 3 x }$ at a = 0.

Accepted Answer

A) 1 + x 
B) 1 + 3x 
C) 1 - x 
D) 1 - 3x 
E) 3 + 3x
F)-3x
G)3 - 3x
H)None of these 
A) 1 + x 
B) 1 + 3x 
C) 1 - x 
D) 1 - 3x 
E) 3 + 3x
F)-3x
G)3 - 3x
H)None of these

Question 9

Suppose that h (x) = f (g (x)) and that we are given the following information: $$\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline x & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 \
\hline f ( x ) & 2.2 & 2.4 & 2.7 & 3.0 & 3.3 & 3.6 & 4.0 \
\hline g ( x ) & 0.85 & 0.60 & 0.50 & 0.40 & 0.25 & 0.10 & 0.00 \
\hline
\end{array}$$ Use the table to estimate the value of $h ^ { \prime }$ (0:3). Justify your estimation.

Accepted Answer

The answer of Suppose that h (x) = f (g...

Question 10

$\begin{array} { | r | c | c | c | c | } 
\hline x & f ( x ) & f ^ { \prime } ( x ) & g ( x ) & g ^ { \prime } ( x ) \
\hline - 6 & 7 & - 8 & - 6 & 7 \
- 4 & 1 & - 5 & 0 & 5 \
- 2 & 3 & - 2 & 4 & 3 \
0 & 5 & 0 & 6 & 1 \
2 & - 5 & 1 & 6 & - 1 \
4 & - 3 & 3 & 4 & - 3 \
6 & 1 & 5 & 0 & - 5 \
\hline
\end{array}$
-Find $\frac { d } { d x } \left( \frac { f ( x ) } { g ( x ) } \right) \text { when } x = 0$

Accepted Answer

A)  $- \frac { 7 } { 36 }$ 
B)  $- \frac { 5 } { 36 }$ 
C)  $- \frac { 3 } { 36 }$ 
D)  $- \frac { 1 } { 36 }$ 
E)  $\frac { 3 } { 36 }$
F) $\frac { 5 } { 36 }$

G) $\frac { 7 } { 36 }$

H) $\frac { 11 } { 36 }$ 
A)  $- \frac { 7 } { 36 }$ 
B)  $- \frac { 5 } { 36 }$ 
C)  $- \frac { 3 } { 36 }$ 
D)  $- \frac { 1 } { 36 }$ 
E)  $\frac { 3 } { 36 }$
F) $\frac { 5 } { 36 }$

G) $\frac { 7 } { 36 }$

H) $\frac { 11 } { 36 }$

Question 11

Find the critical numbers for $f ( x ) = \tan ^ { - 1 } \left( \frac { x } { a } \right) - \tan ^ { - 1 } \left( \frac { x } { b } \right) , a > b$ and identify each as a relative maximum, relative minimum, or neither.

Accepted Answer

@#IMG-DLM& If @#IMG-DLM& does not have critical nu

Question 12

Let $f ( x ) = \sqrt { x } \ln x$ . Find the interval on which $f$ is increasing.

Accepted Answer

A)  $( 0 , \infty )$ 
B)  $( 0,1 ]$ 
C)  $[ 1 , \infty )$ 
D)  $\left( 0 , e ^ { - 1 } \right]$ 
E)  $\left[ e ^ { - 1 } , \infty \right)$ 
F) $\left( 0 , e ^ { - 2 } \right]$ 
G) $\left[ e ^ { - 2 } , \infty \right)$
H) $( 0 , \sqrt { e } )$ 
A)  $( 0 , \infty )$ 
B)  $( 0,1 ]$ 
C)  $[ 1 , \infty )$ 
D)  $\left( 0 , e ^ { - 1 } \right]$ 
E)  $\left[ e ^ { - 1 } , \infty \right)$ 
F) $\left( 0 , e ^ { - 2 } \right]$ 
G) $\left[ e ^ { - 2 } , \infty \right)$
H) $( 0 , \sqrt { e } )$

Question 13

Passing through the origin (0, 0), there are two lines tangent to the curve $y = x ^ { 2 } + 1$ , one with negative slope, the other with positive slope. Find the value of the positive slope.

Accepted Answer

A)  $\frac { 1 } { 8 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 3 }$ 
E) 4
F)1
G)3
H)2 
A)  $\frac { 1 } { 8 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 3 }$ 
E) 4
F)1
G)3
H)2

Question 14

Show that the curves $x ^ { 3 } - 3 x y ^ { 2 } + y = 0$ and $3 x ^ { 2 } y - y ^ { 3 } - x = 4$ are orthogonal.

Accepted Answer

@#IMG-DLM& @#IMG-DLM& , 3x²y -

Question 15

Find the point(s) where the tangent to the curve $y = \frac { x } { x ^ { 2 } + 4 }$ has zero slope.

Accepted Answer

The answer of Find the point(s) where the tangent to...

Question 16

Find the linear approximation of the function f (x) = $\sqrt { x + 24 }$ at x $1$ = 1 and use it to approximate $\sqrt { 24.98 }$ .

Accepted Answer

A) 4.888 
B) 4.948 
C) 4.958 
D) 4.968 
E) 4.978
F)4.988
G)4.998
H)4.9995 
A) 4.888 
B) 4.948 
C) 4.958 
D) 4.968 
E) 4.978
F)4.988
G)4.998
H)4.9995

Question 17

If $x ^ { 4 } + y ^ { 4 } = 1$ , find an expression for $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ .

Accepted Answer

A)  $\frac { y ^ { 2 } } { x ^ { 5 } }$ 
B) 
$\frac { - x ^ { 2 } } { y ^ { 2 } }$ 
C)  $\frac { - 2 x ^ { 4 } } { y ^ { 4 } }$ 
D)  $\frac { - 2 x ^ { 2 } } { y ^ { 5 } }$ 
E)  $\frac { - 3 x ^ { 2 } } { y ^ { 7 } }$ 
F) $\frac { - 5 x ^ { 4 } } { y ^ { 9 } }$ 
G) $\frac { - 6 x ^ { 4 } } { y ^ { 5 } }$ 
H) $\frac { 6 x ^ { 4 } } { y ^ { 7 } }$ 
A)  $\frac { y ^ { 2 } } { x ^ { 5 } }$ 
B) 
$\frac { - x ^ { 2 } } { y ^ { 2 } }$ 
C)  $\frac { - 2 x ^ { 4 } } { y ^ { 4 } }$ 
D)  $\frac { - 2 x ^ { 2 } } { y ^ { 5 } }$ 
E)  $\frac { - 3 x ^ { 2 } } { y ^ { 7 } }$ 
F) $\frac { - 5 x ^ { 4 } } { y ^ { 9 } }$ 
G) $\frac { - 6 x ^ { 4 } } { y ^ { 5 } }$ 
H) $\frac { 6 x ^ { 4 } } { y ^ { 7 } }$

Question 18

The following table shows the relationship between pressure (in atmospheres) and volume (in liters) of hydrogen gas at 0 °C. $$\begin{array} { | l | c | c | c | c | c | c | } 
\hline \text { Pressure (atm) } & 1 & 2 & 3 & 4 & 5 & 6 \
\hline \text { Volume (L) } & 22.4 & 11.2 & 7.5 & 5.6 & 4.5 & 3.7 \
\hline
\end{array}$$ (a) Find the average rate of change of volume with respect to pressure for the following pressure intervals:
(i) [1, 3]
(ii) [2, 3]
(iii) [4, 5]
(b) Plot the data points and fit an appropriate power function to these data.(c) Use the model from part (b) and determine the instantaneous rate of change of volume with respect to pressure.(d) Compare the instantaneous rate at P = 5 with the average rate for [4, 5]. Which is larger? Why is this the case?

Accepted Answer

(a) (i) -7.45 L/atm
(ii) -3.7

Question 19

Find the derivative of $f ( x ) = ( x - \sqrt { x } ) ( x + \sqrt { x } )$ .

Accepted Answer

A)  $\left( 1 - \frac { 1 } { \sqrt { x } } \right) \left( 1 + \frac { 1 } { \sqrt { x } } \right)$ 
B)  $2 x - 1$ 
C)  $\left( 1 - \frac { 1 } { 2 \sqrt { x } } \right) \left( 1 + \frac { 1 } { 2 \sqrt { x } } \right)$ 
D)  $2 x + 1$ 
E)  $\left( 1 - \frac { 1 } { 2 \sqrt { x } } \right) \left( 1 + \frac { 1 } { \sqrt { x } } \right)$
F) $x - 1$

G) $\left( 1 - \frac { 1 } { \sqrt { x } } \right) \left( 1 + \frac { 1 } { 2 \sqrt { x } } \right)$

H)None of these 
A)  $\left( 1 - \frac { 1 } { \sqrt { x } } \right) \left( 1 + \frac { 1 } { \sqrt { x } } \right)$ 
B)  $2 x - 1$ 
C)  $\left( 1 - \frac { 1 } { 2 \sqrt { x } } \right) \left( 1 + \frac { 1 } { 2 \sqrt { x } } \right)$ 
D)  $2 x + 1$ 
E)  $\left( 1 - \frac { 1 } { 2 \sqrt { x } } \right) \left( 1 + \frac { 1 } { \sqrt { x } } \right)$
F) $x - 1$

G) $\left( 1 - \frac { 1 } { \sqrt { x } } \right) \left( 1 + \frac { 1 } { 2 \sqrt { x } } \right)$

H)None of these

Question 20

Find the slope of the tangent to the curve $y = \sin ^ { 2 } x + \sin 2 x$ when $x = \frac { \pi } { 4 }$ .

Accepted Answer

A)  $- \frac { 1 } { 4 }$ 
B)  $- \frac { \sqrt { 2 } } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D) 0 
E)  $\frac { 1 } { 2 }$  
F) $\frac { \sqrt { 2 } } { 2 }$
G)1

H) $\frac { 3 } { 2 }$ 
A)  $- \frac { 1 } { 4 }$ 
B)  $- \frac { \sqrt { 2 } } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D) 0 
E)  $\frac { 1 } { 2 }$  
F) $\frac { \sqrt { 2 } } { 2 }$
G)1

H) $\frac { 3 } { 2 }$

Find an equation of the tangent line to the graph of $y = \frac { \ln x } { x }$ at the point $\left( e ^ { 2 } , \frac { 2 } { e ^ { 2 } } \right)$ .

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for the parametric curve given by $x = t ^ { 4 } - t ^ { 2 } + t , y = \sqrt [ 3 ] { t }$

If $f ( x ) = x ^ { 5 }$ , then, $f ^ { ( 5 ) } ( 3 )$ .

Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 2 x }$ .

Find $\frac { d y } { d x }$ .(a) $y = \left( \frac { x + 1 } { x - 3 } \right) ^ { 4 }$ (b) $y = e ^ { 2 x } + e ^ { 99 }$ (c) $y = x e ^ { x ^ { 2 } + 7 }$ (d) $y = \tan \left( \sin x ^ { 2 } \right)$

Find the exact value of $\sin ^ { - 1 } \left( \sin \frac { 5 \pi } { 6 } \right)$ .

Find the linear approximation to $f ( x ) = e ^ { - 3 x }$ at a = 0.

Suppose that h (x) = f (g (x)) and that we are given the following information: x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 f(x) 2.2 2.4 2.7 3.0 3.3 3.6 4.0 g(x) 0.85 0.60 0.50 0.40 0.25 0.10 0.00 Use the table to estimate the value of $h ^ { \prime }$ (0:3). Justify your estimation.

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } \left( \frac { f ( x ) } { g ( x ) } \right) \text { when } x = 0$

Find the critical numbers for $f ( x ) = \tan ^ { - 1 } \left( \frac { x } { a } \right) - \tan ^ { - 1 } \left( \frac { x } { b } \right) , a > b$ and identify each as a relative maximum, relative minimum, or neither.

Let $f ( x ) = \sqrt { x } \ln x$ . Find the interval on which $f$ is increasing.

Passing through the origin (0, 0), there are two lines tangent to the curve $y = x ^ { 2 } + 1$ , one with negative slope, the other with positive slope. Find the value of the positive slope.

Show that the curves $x ^ { 3 } - 3 x y ^ { 2 } + y = 0$ and $3 x ^ { 2 } y - y ^ { 3 } - x = 4$ are orthogonal.

Find the point(s) where the tangent to the curve $y = \frac { x } { x ^ { 2 } + 4 }$ has zero slope.

Find the linear approximation of the function f (x) = $\sqrt { x + 24 }$ at x $1$ = 1 and use it to approximate $\sqrt { 24.98 }$ .

If $x ^ { 4 } + y ^ { 4 } = 1$ , find an expression for $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ .

Find the derivative of $f ( x ) = ( x - \sqrt { x } ) ( x + \sqrt { x } )$ .

Find the slope of the tangent to the curve $y = \sin ^ { 2 } x + \sin 2 x$ when $x = \frac { \pi } { 4 }$ .

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Exam 3: Differentiation Rules

Find an equation of the tangent line to the graph of y=ln⁡xxy = \frac { \ln x } { x }y=xlnx​ at the point (e2,2e2)\left( e ^ { 2 } , \frac { 2 } { e ^ { 2 } } \right)(e2,e22​) .

Find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ for the parametric curve given by x=t4−t2+t,y=t3x = t ^ { 4 } - t ^ { 2 } + t , y = \sqrt [ 3 ] { t }x=t4−t2+t,y=3t​

If f(x)=x5f ( x ) = x ^ { 5 }f(x)=x5 , then, f(5)(3)f ^ { ( 5 ) } ( 3 )f(5)(3) .

Find the value of the limit lim⁡x→0cos⁡x−12x\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 2 x }limx→0​2xcosx−1​ .

Find dydx\frac { d y } { d x }dxdy​ .(a) y=(x+1x−3)4y = \left( \frac { x + 1 } { x - 3 } \right) ^ { 4 }y=(x−3x+1​)4 (b) y=e2x+e99y = e ^ { 2 x } + e ^ { 99 }y=e2x+e99 (c) y=xex2+7y = x e ^ { x ^ { 2 } + 7 }y=xex2+7 (d) y=tan⁡(sin⁡x2)y = \tan \left( \sin x ^ { 2 } \right)y=tan(sinx2)

Find the exact value of sin⁡−1(sin⁡5π6)\sin ^ { - 1 } \left( \sin \frac { 5 \pi } { 6 } \right)sin−1(sin65π​) .

Find the linear approximation to f(x)=e−3xf ( x ) = e ^ { - 3 x }f(x)=e−3x at a = 0.

Suppose that h (x) = f (g (x)) and that we are given the following information: x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 f(x) 2.2 2.4 2.7 3.0 3.3 3.6 4.0 g(x) 0.85 0.60 0.50 0.40 0.25 0.10 0.00 Use the table to estimate the value of h′h ^ { \prime }h′ (0:3). Justify your estimation.

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find ddx(f(x)g(x)) when x=0\frac { d } { d x } \left( \frac { f ( x ) } { g ( x ) } \right) \text { when } x = 0dxd​(g(x)f(x)​) when x=0

Find the critical numbers for f(x)=tan⁡−1(xa)−tan⁡−1(xb),a>bf ( x ) = \tan ^ { - 1 } \left( \frac { x } { a } \right) - \tan ^ { - 1 } \left( \frac { x } { b } \right) , a > bf(x)=tan−1(ax​)−tan−1(bx​),a>b and identify each as a relative maximum, relative minimum, or neither.

Let f(x)=xln⁡xf ( x ) = \sqrt { x } \ln xf(x)=x​lnx . Find the interval on which fff is increasing.

Passing through the origin (0, 0), there are two lines tangent to the curve y=x2+1y = x ^ { 2 } + 1y=x2+1 , one with negative slope, the other with positive slope. Find the value of the positive slope.

Show that the curves x3−3xy2+y=0x ^ { 3 } - 3 x y ^ { 2 } + y = 0x3−3xy2+y=0 and 3x2y−y3−x=43 x ^ { 2 } y - y ^ { 3 } - x = 43x2y−y3−x=4 are orthogonal.

Find the point(s) where the tangent to the curve y=xx2+4y = \frac { x } { x ^ { 2 } + 4 }y=x2+4x​ has zero slope.

Find the linear approximation of the function f (x) = x+24\sqrt { x + 24 }x+24​ at x 111 = 1 and use it to approximate 24.98\sqrt { 24.98 }24.98​ .

If x4+y4=1x ^ { 4 } + y ^ { 4 } = 1x4+y4=1 , find an expression for d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ .

Find the derivative of f(x)=(x−x)(x+x)f ( x ) = ( x - \sqrt { x } ) ( x + \sqrt { x } )f(x)=(x−x​)(x+x​) .

Find the slope of the tangent to the curve y=sin⁡2x+sin⁡2xy = \sin ^ { 2 } x + \sin 2 xy=sin2x+sin2x when x=π4x = \frac { \pi } { 4 }x=4π​ .

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Find an equation of the tangent line to the graph of $y = \frac { \ln x } { x }$ at the point $\left( e ^ { 2 } , \frac { 2 } { e ^ { 2 } } \right)$ .

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for the parametric curve given by $x = t ^ { 4 } - t ^ { 2 } + t , y = \sqrt [ 3 ] { t }$

If $f ( x ) = x ^ { 5 }$ , then, $f ^ { ( 5 ) } ( 3 )$ .

Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 2 x }$ .

Find $\frac { d y } { d x }$ .(a) $y = \left( \frac { x + 1 } { x - 3 } \right) ^ { 4 }$ (b) $y = e ^ { 2 x } + e ^ { 99 }$ (c) $y = x e ^ { x ^ { 2 } + 7 }$ (d) $y = \tan \left( \sin x ^ { 2 } \right)$

Find the exact value of $\sin ^ { - 1 } \left( \sin \frac { 5 \pi } { 6 } \right)$ .

Find the linear approximation to $f ( x ) = e ^ { - 3 x }$ at a = 0.

Suppose that h (x) = f (g (x)) and that we are given the following information: x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 f(x) 2.2 2.4 2.7 3.0 3.3 3.6 4.0 g(x) 0.85 0.60 0.50 0.40 0.25 0.10 0.00 Use the table to estimate the value of $h ^ { \prime }$ (0:3). Justify your estimation.

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } \left( \frac { f ( x ) } { g ( x ) } \right) \text { when } x = 0$

Find the critical numbers for $f ( x ) = \tan ^ { - 1 } \left( \frac { x } { a } \right) - \tan ^ { - 1 } \left( \frac { x } { b } \right) , a > b$ and identify each as a relative maximum, relative minimum, or neither.

Let $f ( x ) = \sqrt { x } \ln x$ . Find the interval on which $f$ is increasing.

Passing through the origin (0, 0), there are two lines tangent to the curve $y = x ^ { 2 } + 1$ , one with negative slope, the other with positive slope. Find the value of the positive slope.

Show that the curves $x ^ { 3 } - 3 x y ^ { 2 } + y = 0$ and $3 x ^ { 2 } y - y ^ { 3 } - x = 4$ are orthogonal.

Find the point(s) where the tangent to the curve $y = \frac { x } { x ^ { 2 } + 4 }$ has zero slope.

Find the linear approximation of the function f (x) = $\sqrt { x + 24 }$ at x $1$ = 1 and use it to approximate $\sqrt { 24.98 }$ .

If $x ^ { 4 } + y ^ { 4 } = 1$ , find an expression for $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ .

Find the derivative of $f ( x ) = ( x - \sqrt { x } ) ( x + \sqrt { x } )$ .

Find the slope of the tangent to the curve $y = \sin ^ { 2 } x + \sin 2 x$ when $x = \frac { \pi } { 4 }$ .