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The graphs of f and g are shown. Let h(x) = f(g(x)) and s(x) = g(J(x)).
Find each derivative , if it exists. If the derivative ve does not exist, explain why.
Find h'( 1).

Yareli Bowman 2022-08-18

The graphs of f and g are shown. Let h(x) = f(g(x)) and s(x) = g(J(x)).
Find each derivative , if it exists. If the derivative ve does not exist, explain why.
Find s'(5).

Kade Rosales 2022-08-17

Prove $lim_{x \to 0} \sqrt{4 - x} = 2$ using the precise definition of limits.

Leypoldon 2022-08-17

Find $f^{'} (x)$
$f (x) = \frac{1}{\sqrt[3]{x^{5} + x + 3}}$

pleitatsj1 2022-08-16

For the number
$z = i^{i^{i}}$
can I take a log on both sides and write it as $\log (z) = i^{i} \log (i)$ ?
I know that we can write $\log (e^{i θ}) = i θ \log (e)$ , but I'm no sure if we can do that when the base is non real. Is this step valid for a non real base?

jiawu74 2022-08-14

Bierlehre59 2022-08-14

Do the logarithmic rules work when taking logs of functions as opposed to numbers?
i.e. suppose f is a function and n is a real number, is $\log (f (x)^{n}) = n \cdot \log (f (x))$ ?

Lacey Rojas 2022-08-13

How do you find the derivative of $\frac{1}{x}$ ?
Thanks.

Carsen Patel 2022-08-12

Show that the equation $x^{3} + 1 = 15 x$ has three solutions in the interval $[- 4, 4]$

sondestiny120g 2022-08-11

We know that:
$e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$
$\log (x) = \sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1} (x - 1)^{n}}{n}$
Where second series converges when |x−1|<1.It is possible to prove that:
$e^{\log (x)} = x$
for $| x - 1 | < 1$ using only series representation?

logosdepmpe 2022-08-11

use the laws of logarithms to expand the expression
$\log_{3} (\frac{x (x^{2} + 5)}{\sqrt{x^{2} - 5}})$

joyoshibb 2022-08-10

Verifying Fourier series of antiderivative of a function
The function is $f (x) = {\begin{cases} 0, & - π \leq x \leq 0 \\ x, & 0 < x \leq π \end{cases}$ .
The Fourier series of the function is:
$\frac{1}{2} (\frac{π}{2}) + \sum_{n = 1}^{\infty} \frac{(- 1)^{n} - 1}{π n^{2}} \cos (n x) + \frac{(- 1)^{n + 1}}{n} \sin (n x)$ .
Since the function can be integrated termwise, the Fourier series for the antiderivative $\int_{- π}^{x} f (x) d x$ is:
$\frac{1}{2} (\frac{π}{2}) (x + π) + \sum_{n = 1}^{\infty} \frac{1}{n} (\frac{(- 1)^{n} - 1}{π n^{2}} \sin (n x) + \frac{(- 1)^{n}}{n} (\cos (n x) - (- 1)^{n}))$
The antiderivative of the function is $F (x) = {\begin{cases} 0, & - π \leq x \leq 0 \\ \frac{x^{2}}{2}, & 0 < x \leq π \end{cases}$ . The Fourier series of the antiderivative is:
$\frac{1}{2} (\frac{π^{2}}{6}) + \sum_{n = 1}^{\infty} \frac{1}{n} ((\frac{(- 1)^{n} - 1}{π n^{2}} - (- 1)^{n} \frac{π}{2}) \sin (n x) + \frac{(- 1)^{n}}{n} \cos (n x))$
Where I am missing? Why I am having wrong answer. Even if I did some mistakes in calculations, what's with x term obtained in piece wise integration. Thanks in advance.

Garrett Sheppard 2022-08-10

I'm confused trying to figure out the domain of the logarithmic function below:
$f (x) = \ln (e^{x} + 3)$
Because the argument of $f, e^{x} + 3,$ , is a nonnegative number, $e^{x} + 3 > 0$ and $e^{x} > - 3.$ Taking natural logarithms on both sides, we get $x > \ln (- 3)$ . However, the domain of a logarithmic function must be nonnegative real numbers, so $\ln (- 3)$ doesn't make sense. How then to determine the scope of the original function?

granuliet1u 2022-08-10

I have the following function and I am trying to isolate x:
$\ln (x) + \ln (x - 1) = 1$
I take ex of both sides:
$x + x - 1 = e$
$2 x - 1 = e$
$2 x = e + 1$
$x = \frac{e + 1}{2}$

Trevor Rush 2022-08-09

Find the solution:
$lim_{x \to \infty} \frac{x e^{x - 1}}{(x - 1) e^{x}}$

empalhaviyt 2022-08-08

Find the limt. $lim_{x \to 0} \frac{(x \csc 2 x)}{\cos 5 x}$

Holzkeulecz 2022-08-08

Evaluate the integral: $\int 4 x \sec^{2} x d x$

Garrett Sheppard 2022-08-08

The minute hand of a clock is 6 inches long and moves from 12 to 4 o'clock. How far does the tip of the minute hand move? Express your answer in terms of pi, then round two decimal places.

Yair Valentine 2022-08-07

Simplify. Write the answer as a single term without negative exponents.
$\frac{7 n^{-} 7 - 7 m^{-} 7}{m + n^{8}}$

Matonya 2022-08-07

Let h(x)=10x and j(x)=ex. Find the following.
j(3.5)

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