Hi
I am struggling to solve the equations:
Given:
d(e^t)/dt =e^t
use the chain rule to find dy/dx:
a) y=e^5t,
b) y=4e^3t
c)y=6e^-2t
(where "^" represents an exponent)
Thanks in advance for any help
Motty
The Chain Rule states,
$\displaystyle \frac{d}{dx}f(g(x))=f'(g(x))g'(x).$
It can be seen to work through the fact that $\displaystyle g(x)$ changes at the rate $\displaystyle g'(x)$ as $\displaystyle x$ moves, speeding up (or slowing down) the rate of change of $\displaystyle f(g(x))$ by that amount.
For instance, if $\displaystyle g'(x)=2$, then $\displaystyle g(x)$ moves twice as fast, and $\displaystyle \frac{d}{dx}f(g(x))=f'(g(x))\cdot 2$.
For problem (a), we note that $\displaystyle f(t)=e^t$ and $\displaystyle g(t)=5t$. The Chain Rule therefore states,
$\displaystyle \frac{d}{dt}(e^{5t})=\frac{d}{dt}f(g(t))=f'(g(t))g '(t)=e^{g(t)}\frac{d}{dt}(5t)=e^{5t}\cdot 5=5e^{5t}.$
The same reasoning may be applied to (b) and (c).