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# how do you do the derivative of 6.687(.931)^x?

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how do you do the derivative of 6.687(.931)^x?

Guest Apr 16, 2015

#2
+94106
+5

$$\boxed{If\;\;y=a^x\;\;then\;\;\frac{dy}{dx}=(lna)a^x}$$

Mmm, I didn't know this short cut Chris,  I will have to try and remember it!    Wish me luck. :)

This is how I would have done it,

$$\\y=6.687*0.9316^x\\\\ \frac{y}{6.687}=0.9316^x\\\\ ln(\frac{y}{6.687})=ln(0.9316^x)\\\\ ln(y)-ln(6.6870)=xln(0.9316)\\\\ \frac{1}{y}\;\frac{dy}{dx}-0=ln(0.9316)\\\\ \frac{dy}{dx}=yln(0.9316)\\\\ sub in y\\\\ \frac{dy}{dx}=6.687*0.9316^x*ln(0.9316)\\\\ \frac{dy}{dx}=6.687*ln(0.9316)*0.9316^x\\\\$$

Melody  Apr 17, 2015
#1
+92699
+5

The derivative of a^x =   ln(a) * a^x

So, the derivative of (.931)^x  =   ln(.931) * (.931)^x

You can evaluate ln(.931) on your calculator......then.....multiply that times 6.687  and then "tack" the   "(.931)^x  "  part on at the end.......

CPhill  Apr 16, 2015
#2
+94106
+5

$$\boxed{If\;\;y=a^x\;\;then\;\;\frac{dy}{dx}=(lna)a^x}$$

Mmm, I didn't know this short cut Chris,  I will have to try and remember it!    Wish me luck. :)

This is how I would have done it,

$$\\y=6.687*0.9316^x\\\\ \frac{y}{6.687}=0.9316^x\\\\ ln(\frac{y}{6.687})=ln(0.9316^x)\\\\ ln(y)-ln(6.6870)=xln(0.9316)\\\\ \frac{1}{y}\;\frac{dy}{dx}-0=ln(0.9316)\\\\ \frac{dy}{dx}=yln(0.9316)\\\\ sub in y\\\\ \frac{dy}{dx}=6.687*0.9316^x*ln(0.9316)\\\\ \frac{dy}{dx}=6.687*ln(0.9316)*0.9316^x\\\\$$

Melody  Apr 17, 2015