# Thread: Chain Rules - Partial Derivatives

1. ## Chain Rules - Partial Derivatives

Let $\displaystyle z =f(x,y) = (8+25x)y^{tan(x)})$ , where: $\displaystyle x = (\frac{12 - e}{2e})(t-1) + \tan^{-1}(t)$, and $\displaystyle y = (2 - t)e^{t} - 12(t-1)$. Use the chain rule to find $\displaystyle \frac{dz}{dt}$ at $\displaystyle t = 1$.

$\displaystyle \frac{dz}{dt} = ?$

The Chain rule formula I used is the following: $\displaystyle \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{\partial y}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$

And, when $\displaystyle t = 1$, $\displaystyle x = \frac{\pi}{4}$
When $\displaystyle t = 1$, $\displaystyle y = e$

$\displaystyle \frac{\partial z}{\partial x} = (8 + 25x)y^{\sec^{2}(x)} + 25y^{\tan(x)}$

$\displaystyle \frac{\partial z}{\partial y} = (8 + 25x)\tan(x)y^{\tan(x) - 1}$. This is one correct?

$\displaystyle \frac{\partial y}{\partial t} = (2 - t)e^{t} + e^{t}$

$\displaystyle \frac{\partial x}{\partial t} = \frac{6}{e} - \frac{1}{2} + \frac{1}{1+ t^{2}}$

2. Originally Posted by Belowzero78
Let $\displaystyle z =f(x,y) = (8+25x)y^{tan(x)})$ , where: $\displaystyle x = (\frac{12 - e}{2e})(t-1) + \tan^{-1}(t)$, and $\displaystyle y = (2 - t)e^{t} - 12(t-1)$. Use the chain rule to find $\displaystyle \frac{dz}{dt}$ at $\displaystyle t = 1$.

$\displaystyle \frac{dz}{dt} = ?$

The Chain rule formula I used is the following: $\displaystyle \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{\partial y}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$
In your first partial, you should an x wrt t not y wrt t but you did the correct derivatives in your work.

And, when $\displaystyle t = 1$, $\displaystyle x = \frac{\pi}{4}$
When $\displaystyle t = 1$, $\displaystyle y = e$

$\displaystyle \frac{\partial z}{\partial x} = (8 + 25x)y^{\sec^{2}(x)} + 25y^{\tan(x)}$

$\displaystyle \frac{\partial z}{\partial y} = (8 + 25x)\tan(x)y^{\tan(x) - 1}$. This is one correct?

$\displaystyle \frac{\partial y}{\partial t} = (2 - t)e^{t} + e^{t}$

$\displaystyle \frac{\partial x}{\partial t} = \frac{6}{e} - \frac{1}{2} + \frac{1}{1+ t^{2}}$
Looks fine besides the typo; however, I didn't check your derivatives. I am little confused on what you are asking. You want to know what $\displaystyle \frac{dz}{dt}$ is but you have done everything as long as your math is correct.

3. Could someone please show me what the partial is for: $\displaystyle \frac{\partial z}{\partial x}$?
I'm really confused about the y^tan(x) part when taking the derivative of x.

Would it be: $\displaystyle \frac{\partial z}{\partial x} = (8 + 25x)y^{\tan(x)}\ln(y)\sec^{2}(x) + 25y^{\tan(x)}$ ?

4. For example, $\displaystyle y^{tan(x)}$ can be solved like this:

$\displaystyle y=u$ and $\displaystyle tanx=v$

$\displaystyle \big[\frac{u'*v}{u}+ln(u)*v'\big]*u^v$

Therefore, $\displaystyle 25y^{tan(x)}=25\frac{y^{tan(x)}*ln(y)}{cos^2(x)}$

6. Add your solutions $\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$ of the partial derivatives you obtain.