Question

Logarithmic Function Derivatives (Calculus)

Answer 1

Alright, so lets look at the formula that we have:

Base e:
\[d/dx(lnx) = 1/x\]

Any Base:
\[d/dx(\log_{a} x)= 1/(x \times \ln(a))\]

Some examples:

\[y=\ln(2x-3)\] find d/dx

Chain rule + Log derivative rule
\[dy/dx = 1/(2x-3) \times 2 = 2/(2x-3)\]

Next example:

\[y=( \log_{5}x^3 )\]
\[dy/dx=1/(x^3 \times \ln(5)) \times 3x^2 = 3x^2/(x^3 \times \ln(5)) = 3/(x \times \ln(5))\]

What we did here is apply the formula and multiply by 3x^2 because of chain rule

Answer 2

Logarithmic Differientation Now we are going to learn about logarithmic differientation,. What is this: this is a strategy to simplify problems that involve a variable raised to a variable, or problems that involve too many derivative rules. If you want to refer on how to do Implicit differentiation, link right below. https://questioncove.com/study#/updates/5fb6d473283658df98d49000 Steps: 1) Take the natural log of both sides of the equation 2) Simplify with log laws 3) Find the derivative implicitly Ill also post the log laws, they are as following \[\log(ab)=\log(a)+\log(b)\] \[\log(a/b)=\log(a)-\log(b)\] \[\log(a^b)= b \log(a)\] \[\log_{x}(1/x^a)=-a \] \[\log_{x}1=0 \]

Answer 3

Lets take a look at one example
\[y=x^{4x}\] find y'

\[\ln(y)=\ln(x ^{4x})\]
Using the power rule of logs, we can rewrite \[\ln(x^{4x}) = 4x \times \ln(x)\]

So now \[\ln(y)=4x \times \ln(x)\]
Now we take the derivative of both sides, for more in-depth explanations on this specific step refer to my Implicit differentiation tutorial. Anyways, the derivative of \[\ln(y)=1/y\] if you remember from our base e rule for logs earlier in this tutorial. But that isn't the full thing because anytime you are finding the derivative with a term y you have to multiply by the derivative of y,
So the derivative of ln(y) is \[(1/y) \times y'\]

Now on the right side of the equation, we have 4x times ln(x), so we need the product rule of derivatives. The product rule of derivatives is:
\[d/dx(f *g)=f *g'+g*f'\]
We just have to assign 4x to f and ln(x) to g to use this formula

Answer 4

\[(1/y)*(y')= 4x *1/x +\ln(x) * 4\]

\[y'=4+4*\ln(x)*y\]
\[y=x ^{4x}\]
\[y'=(4+4(\ln(x))*x ^{4x}\]