1. Trigonometry Function Derivative

Q: $\displaystyle f(x) = \mathrm{arcsin} x, \ -1 \le x \le 1$.

(a)Evaluate $\displaystyle f \left( - \frac{1}{2} \right)$.

(b) Find an equation of the tangent to the curve with equation $\displaystyle y=f(x)$ at the point where $\displaystyle x = \frac{1}{\sqrt{2}}$.

__________________
I've done part (a) but struggling with part (b). May I have some help please.

2. Hi

The equation of the tangent to the curve at a point $\displaystyle (a,\,f(a))$ is $\displaystyle y=f(a)+(x-a)\cdot f'(a)$. In your case, it becomes $\displaystyle y=f\left(\frac{1}{\sqrt{2}}\right)+\left(x-\frac{1}{\sqrt{2}}\right)\cdot f'\left(\frac{1}{\sqrt{2}}\right)=\ldots$

3. But...I still don't get it very well. I understand that the coordinates to consider is $\displaystyle \left( \frac{1}{\sqrt{2}}, \frac{\pi}{2} \right)$. Can someone work out the gradient? I think I might have got that wrong.

4. Yop

Originally Posted by Air
Q: $\displaystyle f(x) = \mathrm{arcsin} x, \ -1 \le x \le 1$.

(a)Evaluate $\displaystyle f \left( - \frac{1}{2} \right)$.

(b) Find an equation of the tangent to the curve with equation $\displaystyle y=f(x)$ at the point where $\displaystyle x = \frac{1}{\sqrt{2}}$.

__________________
I've done part (a) but struggling with part (b). May I have some help please.
The equation of the tangent to a curve whose equation is f(x) is :

$\displaystyle y=f'(a)(x-a)+f(a)$

Here, $\displaystyle a=\frac{1}{\sqrt{2}}$

$\displaystyle f(x)=\arcsin(x) \Longrightarrow f'(x)=\frac{1}{\sqrt{1-x^2}}$

--> tangent :

$\displaystyle y=\frac{1}{\sqrt{1-(\frac{1}{\sqrt{2}})^2}} \cdot (x-\frac{1}{\sqrt{2}})+f(\frac{1}{\sqrt{2}})$

etc..

5. Another way of getting the gradient:

Gradient of Function y = f'(x) is $\displaystyle \frac{dy}{dx}$

$\displaystyle y = arcsin(x)$
$\displaystyle x = sin(y)$
$\displaystyle \frac{dx}{dy} = cos(y)$
$\displaystyle \frac{dy}{dx} = sec(y)$
but y = arcsin(x)

$\displaystyle \frac{dy}{dx} = sec(arcsin(x))$
at $\displaystyle x = \frac{1}{\sqrt2}$
$\displaystyle \frac{dy}{dx} = sec(\frac{\pi}{4})$
= $\displaystyle \sqrt{2}$