Finding Derivative at a Point Flashcards - High School Math

We need to find the first derivative of f(x). This will require us to apply both the Product and Chain Rules. When we apply the Product Rule, we obtain:

$f'(x)=\sin(x^2)\cdot\frac{\mathrm{d} }{\mathrm{d} x}[x^2]+x^2\cdot\frac{\mathrm{d} }{\mathrm{d} x}[\sin(x^2)]$

In order to find the derivative of $\sin(x^2)$ , we will need to employ the Chain Rule.

$\frac{\mathrm{d} }{\mathrm{d} x}[\sin(x^2)]=\cos(x^2)\cdot\frac{\mathrm{d} }{\mathrm{d} x}[x^2]=\cos(x^2)\cdot2x$

$f'(x)=\sin(x^2)\cdot2x + x^2\cdot\cos(x^2)\cdot2x$

We can factor out a 2x to make this a little nicer to look at.

$f'(x)=2x(\sin(x^2)+x^2\cos(x^2))$

Now we must evaluate the derivative when x = $\frac{\sqrt{\pi}}{2}$ .

$f'\left (\frac{\sqrt{\pi}}{2} \right )=2\cdot\frac{\sqrt{\pi}}{2}(\sin\frac{\pi}{4}+\frac{\pi}{4}\cos\frac{\pi}{4})$

$=\sqrt{\pi}(\frac{\sqrt{2}}{2}+\frac{\pi\sqrt{2}}{8})=\sqrt{2\pi}(\frac{1}{2}+\frac{\pi}{8})=\frac{\sqrt{2\pi}(\pi + 4)}{8}$

The answer is $\frac{\sqrt{2\pi}(\pi + 4)}{8}$ .

Finding Derivative at a Point - High School Math

Find $f^{'}(2)$ if the function is given by

To find the derivative at , we first take the derivative of . By the derivative rule for logarithms,

$f'(x) = \frac{1}{x}$

Plugging in , we get

$f'(2) = \frac{1}{2}$

Find the derivative of the following function at the point .

Use the power rule on each term of the polynomial to get the derivative,

Now we plug in

Let $f(x)=x^2\sin(x^2)$ . What is $f'\left (\frac{\sqrt{\pi}}{2} \right )$ ?

Finding Derivative at a Point - High School Math

Find if the function is given by

To find the derivative at , we first take the derivative of . By the derivative rule for logarithms, Plugging in , we get

Find the derivative of the following function at the point .

Use the power rule on each term of the polynomial to get the derivative, Now we plug in

Let . What is ?

Find $f^{'}(2)$ if the function is given by

To find the derivative at , we first take the derivative of . By the derivative rule for logarithms,

$f'(x) = \frac{1}{x}$

Plugging in , we get

$f'(2) = \frac{1}{2}$

Use the power rule on each term of the polynomial to get the derivative,

Now we plug in

Let $f(x)=x^2\sin(x^2)$ . What is $f'\left (\frac{\sqrt{\pi}}{2} \right )$ ?