Calculate the derivative of the following functions.

y = e^2x(2x-7)⁵

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

y² = x²(4 − x) / 4 + x (right strophoid)

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

y² = x²(4 − x) / 4 + x (right strophoid)

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

y² = x²(4 − x) / 4 + x (right strophoid)

Find f′(x), f′′(x), and f′′′(x) for the following functions.

f(x) = (x² - 7x - 8) / (x + 1)

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{\prime }\left(a\right)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$ and $f_{-}^{\prime }\left(a\right)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$, respectively, provided these limits exist. The derivative $f^{\prime }\left(a\right)$ exists if and only if $f_{+}^{\prime }\left(a\right)=f_{-}^{\prime }\left(a\right)$.

Compute $f_{+}^{\prime }\left(a\right)$ and $f_{-}^{\prime }\left(a\right)$ at the given point $a$.

$f\left(x\right)=\mid x-2\mid$; $a=2$

First and second derivatives Find f′(x),f′′(x).

f(x) = x/x+2

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{\prime }\left(a\right)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$ and $f_{-}^{\prime }\left(a\right)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$, respectively, provided these limits exist. The derivative $f^{\prime}\left(a\right)$ exists if and only if $f_{+}^{\prime}\left(a\right)=f_{-}^{\prime}\left(a\right)$.

Compute $f_{+}^{\prime}\left(a\right)$ and $f_{-}^{\prime}\left(a\right)$ at the given point $a$.

$f\left(x\right)=\{\begin{array}{c}4-x^2\text{ if }x\le 1\\ 2x+1\text{ if }x>1\end{array}$; $a=1$

Calculate the derivative of the following functions.

y = (p+3)² sin p²

Find an equation of the line tangent to the following curves at the given value of x.

y = 1+2 sin x; x = π/6

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = g(x) at x = 3.

Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) and h(x) = f(x) / x−3.

b. Find an equation of the line tangent to y = h(x) at x=2.

27–76. Calculate the derivative of the following functions.

y = √f(x), where f is differentiable and nonnegative at x.

{Use of Tech} The Witch of Agnesi The graph of y = a³ / (x² + a²) , where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

Let a = 3 and find an equation of the line tangent to y = 27 / (x² + 9) at x = 2.

{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a² , where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

b. Plot the function and the tangent line found in part (a).

Derivatives from a table Use the following table to find the given derivatives. <IMAGE>

d/dx (f(x)g(x)) |x=1

Graph the function $f\left(x\right)=\{\begin{array}{c}x\text{ if }x\le 0\\ x+1\text{ if }x>0\end{array}$.

For x < 0, what is f′(x)?

For x > 0, what is f′(x)?

Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives. <IMAGE>

F'(2)

Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives.

<IMAGE>

G'(2)

Derivatives from a table Use the following table to find the given derivatives. <IMAGE>

d/dx (xf(x) / g(x)) |x=4

{Use of Tech} Cell population The ​population ​of a culture of cells after t days is approximated by the function P(t)=1600 / 1 + 7e^−0.02t, for t≥0.

a. Graph the population function.  

The following limits represent f'(a) for some function f and some real number a.

Find a possible function f and number a.

lim x🠂0 e^x-1 / x

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂0 e^x-1 / x

{Use of Tech} Cell population The ​population ​of a culture of cells after t days is approximated by the function P(t)=1600 / 1 + 7e^−0.02t, for t≥0.

e. Graph the growth rate. When is it a maximum and what is the population at the time that the growth rate is a maximum? 

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (5f(x)+3g(x)) |x=1

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂1 x¹⁰⁰-1 / x-1

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (f(f(x))) |x=4

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (g(f(x))) |x=1