Logarithmic differentiation

Hi guys, I am currently finishing my Calculus 1 course and got stuck in this last exercise. I couldn`t find any similar examples on google and tried to do it in many ways and had no success.

$\displaystyle \text{Let } f(x) = (x^4 + 7)^{ln(x)}$

$\displaystyle \text{Find }f^\prime(x) = $

I dont know what to do. I guess I have to put the ln (x) in front. Ive already tried putting ln in both sides but for some reason its also wrong... What am I suppose to do with the exponent ln (x)?

Thanks guys!

Re: Logarithmic differentiation

Quote:

Originally Posted by

**lguto** Hi guys, I am currently finishing my Calculus 1 course and got stuck in this last exercise. I couldn`t find any similar examples on google and tried to do it in many ways and had no success.

$\displaystyle \text{Let } f(x) = (x^4 + 7)^{ln(x)}$

$\displaystyle \text{Find }f^\prime(x) = $

I dont know what to do. I guess I have to put the ln (x) in front. Ive already tried putting ln in both sides but for some reason its also wrong... What am I suppose to do with the exponent ln (x)?

Thanks guys!

$\displaystyle \displaystyle \begin{align*} f &= \left( x^4 + 7 \right) ^{\ln{x}} \\ \ln{(f)} &= \ln{\left[ \left( x^4 + 7 \right)^{\ln{(x)}} \right]} \\ \ln{(f)} &= \ln{(x)}\ln{\left( x^4 + 7 \right)} \\ \frac{d}{dx} \left[ \ln{(f)} \right] &= \frac{d}{dx} \left[ \ln{(x)} \ln{\left( x^4 + 7 \right)} \right] \end{align*}$

See if you can finish it. Of course, on the LHS you will need to use Implicit Differentiation :)

Re: Logarithmic differentiation

Quote:

Originally Posted by

**Prove It** $\displaystyle \displaystyle \begin{align*} f &= \left( x^4 + 7 \right) ^{\ln{x}} \\ \ln{(f)} &= \ln{\left[ \left( x^4 + 7 \right)^{\ln{(x)}} \right]} \\ \ln{(f)} &= \ln{(x)}\ln{\left( x^4 + 7 \right)} \\ \frac{d}{dx} \left[ \ln{(f)} \right] &= \frac{d}{dx} \left[ \ln{(x)} \ln{\left( x^4 + 7 \right)} \right] \end{align*}$

See if you can finish it. Of course, on the LHS you will need to use Implicit Differentiation :)

Thanks a lot for the help!

Well, I tried doing this method and I got it wrong. Please let me know if I made a mistake. Ill continue from where you started:

dy/dx = 1/x * 1/(x^4 + 7) = 1/x^5 + 7x

d/dx = y (1/x^5 + 7x)

Thats what i did and I still got it wrong... I REALLY dot know what m I doing wrong as Im trying to finish this exercise for hours... Maybe it is someting in the algebra? I really dont know :(

Thanks a lot mate!

Re: Logarithmic differentiation

Quote:

Originally Posted by

**lguto** Thanks a lot for the help!

Well, I tried doing this method and I got it wrong. Please let me know if I made a mistake. Ill continue from where you started:

dy/dx = 1/x * 1/(x^4 + 7) = 1/x^5 + 7x

d/dx = y (1/x^5 + 7x)

Thats what i did and I still got it wrong... I REALLY dot know what m I doing wrong as Im trying to finish this exercise for hours... Maybe it is someting in the algebra? I really dont know :(

Thanks a lot mate!

First of all, I don't know where you have pulled a y from...

The RHS is a product of functions, so you need to use the Product Rule to differentiate. One of the terms also requires using the Chain Rule.

Since f is a function of x, it's a composition of functions, so you need to use the Chain Rule to differentiate the LHS.

Re: Logarithmic differentiation

Quote:

Originally Posted by

**Prove It** First of all, I don't know where you have pulled a y from...

The RHS is a product of functions, so you need to use the Product Rule to differentiate. One of the terms also requires using the Chain Rule.

Since f is a function of x, it's a composition of functions, so you need to use the Chain Rule to differentiate the LHS.

Thanks a lot mate! I managed to get it right! I forgot about the chain rule. The implicit differentiation part was right, my mistakes were just basic stuff :(

Thanks a lot!