# Math Help - Logramethic differentaion

1. ## Logramethic differentaion

Determine the question of the tangent line to y = x^sinx AT x = pi/2

This seemed alot easier when looking at it

2. Originally Posted by proski117

Determine the question of the tangent line to y = x^sinx AT x = pi/2

This seemed alot easier when looking at it
$\ln(y) = \ln(x^{\sin(x)})$

$\ln(y) = \sin(x) \ln(x)$

$\frac{d}{dx} \ln(y) = \frac{d}{dx} \sin(x) \ln(x)$

$\frac{dy}{dx} \times \frac{d}{dy} \ln(y) = \frac{d}{dx} \sin(x) \ln(x)$

$\frac{dy}{dx} \times \frac{1}{y} = \frac{d}{dx} \sin(x) \ln(x)$

Now use the product rule on the RHS, substitude the original equation in for 'y', and rearrange to get $\frac{dy}{dx} =$.

3. Originally Posted by proski117

Determine the question of the tangent line to y = x^sinx AT x = pi/2

This seemed alot easier when looking at it
$y = x^{\sin x} \Rightarrow \ln y = \sin x \ln x$.

The derivative of the left hand side is $\frac{1}{y} \cdot \frac{dy}{dx}$. Use the product rule to differentiate the right hand side.

Substitute $x = \frac{\pi}{2}$ and $y = \frac{\pi}{2}$ (why?) and solve for the value of $m = \frac{dy}{dx}$.

You now have the gradient of the tangent and you know a point on the tangent. Getting the equation of the tangent should be routine.

4. Originally Posted by proski117

Determine the question of the tangent line to y = x^sinx AT x = pi/2

This seemed alot easier when looking at it
logarithmic derivative ...

$y = x^{\sin{x}}$

$\ln{y} = \ln\left(x^{\sin{x}}\right)$

$\ln{y} = \sin{x} \cdot \ln{x}$

take the derivative of both sides ...

$\frac{y'}{y} = \sin{x} \cdot \frac{1}{x} + \cos{x} \cdot \ln{x}$

$y' = y\left(\sin{x} \cdot \frac{1}{x} + \cos{x} \cdot \ln{x}\right)$

$y' = x^{\sin{x}}\left(\sin{x} \cdot \frac{1}{x} + \cos{x} \cdot \ln{x}\right)
$

sub in $\frac{\pi}{2}$ for x and evaluate.

5. The question said do not simplify my answer...but when you guys showed me the derivative, i subbed in pi/2 and got 1 as my answer for slope. Did I sub in wrong?

6. Originally Posted by proski117
The question said do not simplify my answer...but when you guys showed me the derivative, i subbed in pi/2 and got 1 as my answer for slope. Did I sub in wrong?
yes, the slope is 1 at that point