$f(x)={\int}_{0}^{\mathrm{sin}x}1+\mathrm{sin}(\mathrm{sin}(t))dt$

Find $({f}^{-1}{)}^{\prime}(0)$

Find $({f}^{-1}{)}^{\prime}(0)$

Emanuel Keith
2022-06-25
Answered

$f(x)={\int}_{0}^{\mathrm{sin}x}1+\mathrm{sin}(\mathrm{sin}(t))dt$

Find $({f}^{-1}{)}^{\prime}(0)$

Find $({f}^{-1}{)}^{\prime}(0)$

You can still ask an expert for help

Blaze Frank

Answered 2022-06-26
Author has **18** answers

$\begin{array}{c}f({f}^{-1}(x))=x\\ {f}^{\prime}({f}^{-1}(x))\ast {({f}^{-1}(x))}^{\prime}=1\\ (from\text{}Chain\text{}Rule)\\ {({f}^{-1}(x))}^{\prime}=\frac{1}{{f}^{\prime}({f}^{-1}(x))}\\ ({f}^{-1}(0{)}^{\prime}=\frac{1}{{f}^{\prime}({f}^{-1}(0))}\\ Now,\text{}from\text{}the\text{}original\text{}eqn,\\ f(x)={\int}_{0}^{sinx}(1+sin(sint))dt\\ {f}^{\prime}(x)=1+sin(sin(sin(x))\ast cosx\\ (from\text{}Leibnitz\text{}formula)\\ Let\text{}{f}^{-1}(0)=y\\ then\text{}f(y)=0,\text{}so\text{}y=0\end{array}$

Just find f'(0) and then you are done.

Just find f'(0) and then you are done.

asked 2022-06-08

Suppose $F(x)={\int}_{3x+8}^{{x}^{2}+5x+1}{\mathrm{csc}}^{2}\left(t\right)dt$. How would one find ${F}^{\prime}(x)$ using the first fundamental theorem of calculus?

asked 2022-06-22

Can use FToC to evaluate $\underset{x\to \mathrm{\infty}}{lim}\frac{{\int}_{0}^{x}\phantom{\rule{mediummathspace}{0ex}}f\left(t\right)dt}{{x}^{2}}$?

asked 2022-06-20

If $f(x)$ is even, then what can we say about:

${\int}_{-2}^{2}f(x)dx$

If $f(x)$ is odd, then what can we say about

${\int}_{-2}^{2}f(x)dx$

Are they both zero? For the first one if its even wouldn't this be the same as

${\int}_{a}^{a}f(x)dx=0$

Now if its odd $f(-x)=-f(x)$. Would FTOC make this zero as well?

${\int}_{-2}^{2}f(x)dx$

If $f(x)$ is odd, then what can we say about

${\int}_{-2}^{2}f(x)dx$

Are they both zero? For the first one if its even wouldn't this be the same as

${\int}_{a}^{a}f(x)dx=0$

Now if its odd $f(-x)=-f(x)$. Would FTOC make this zero as well?

asked 2022-07-01

How do you use the Fundamental Theorem of Calculus to find the derivative of $\int \frac{1}{1+{t}^{2}}dt$ from $x$ to $5$?

asked 2022-05-10

Are there any cases which the First Fundamental Theorem of Calculus would fail?

asked 2022-05-09

First fundamental theorem of calculus uses a function

$F(x)=\text{}{\int}_{a}^{x}f\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$ for $f$ a continuous function between $[a,b]$ where $x$ is between $[a,b]$

$F(x)$ is an antiderivative of function $f$.

So what closed interval is considered when we take indefinite integral?

$F(x)=\text{}{\int}_{a}^{x}f\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$ for $f$ a continuous function between $[a,b]$ where $x$ is between $[a,b]$

$F(x)$ is an antiderivative of function $f$.

So what closed interval is considered when we take indefinite integral?

asked 2022-06-24

Derivative of a function and an integral

$\frac{d}{dx}({x}^{6}({\int}_{0}^{sinx}\sqrt{t}dt))$

$\frac{d}{dx}({x}^{6}({\int}_{0}^{sinx}\sqrt{t}dt))$