# Thread: Differentiate x^(t-1) wrt t

1. ## Differentiate x^(t-1) wrt t

How can we find the partial derivative of $\displaystyle x^{t-1}$ with respect to t? This looks as if it should be really easy, but I'm not seeing it...

2. Originally Posted by Amanda1990
How can we find the partial derivative of $\displaystyle x^{t-1}$ with respect to t? This looks as if it should be really easy, but I'm not seeing it...
Treat $\displaystyle x$ as a constant (let's say $\displaystyle a$). Then it is similar to differentiating $\displaystyle \frac{\,d}{\,du}\left[a^u\right]$.

So it follows that $\displaystyle \frac{\partial}{\partial t}\left[x^{f\!\left(t\right)}\right]=x^{f\!\left(t\right)}\ln x\cdot\frac{\,df}{\,dt}$.

Can you take it from here?

3. Originally Posted by Amanda1990
How can we find the partial derivative of $\displaystyle x^{t-1}$ with respect to t? This looks as if it should be really easy, but I'm not seeing it...
The easiest way is to let

$\displaystyle z = x^{t-1}$ take the natural log $\displaystyle \ln z = (t-1)\ln x$ then take the derivative

Edit. A little late