chain rule

[Janet Ward]

I'm using the chain rule to figure out the problem find the derivative of 4^3^x^2
I labled the outer function to the inner function as follows:
a(t)=x
b(t)=x^2=[a(t)]^2
c(t)=3^x^2=3^[b(t)]
d(t)=4^3^x^2=4^[c(t)]
Now I am trying to find the deivative and I'm confused by the power of a power of a power
do I use the chain rule only or combine it with the power rule how do I get started?
t'=

 

[Deleted member 4993]

I'm using the chain rule to figure out the problem find the derivative of 4^3^x^2
I labled the outer function to the inner function as follows:
a(t)=x
b(t)=x^2=[a(t)]^2
c(t)=3^x^2=3^[b(t)]
d(t)=4^3^x^2=4^[c(t)]
Now I am trying to find the deivative and I'm confused by the power of a power of a power
do I use the chain rule only or combine it with the power rule how do I get started?
t'=

You'll need to log of both sides:

log(t) = c(t) * log(4)

Now continue....

 

[lookagain]

I'm using the chain rule to figure out the problem find the derivative of \(\displaystyle >>\)4^3^x^2\(\displaystyle < <\)

What is this supposed to be?

If you mean \(\displaystyle 4^{3^{x^2}},\)then put in grouping symbols such as:

4^[3^(x^2)]

 

[soroban]

Hello, Janet Ward!

\(\displaystyle \text{I'm using the chain rule to find the derivative of: }\:f(x) \:=\:4^{3^{x^2}}\)

We are expected to know this formula:

. . \(\displaystyle \frac{d}{dx}(a^u) \;=\;a^u\cdot\ln a\cdot\frac{du}{dx}\)

The derivative of an exponential function equals:
. . . (the exponential function) x (natural log of the base) x (derivative of the exponent)


\(\displaystyle \text{So the derivative of }f(x) \:=\:4^{\left(3^{x^2}\right)}\:\text{ equals }\:4^{\left(3^{x^2}\right)} \times \ln4 \times \left(\text{derivative of }3^{x^2}\right)\)

. . \(\displaystyle \text{The derivative of }3^{x^2}\text{ equals: }\:3^{x^2}\times\ln3 \times \left(\text{derivative of }x^2)\,\)

. . . . \(\displaystyle \text{The derivative of }x^2\text{ equals: }\:2x\)


\(\displaystyle \text{Therefore: }\;f'(x) \;=\;\left(4^{3^{x^2}}\!\!\cdot\ln4\right) \left(3^{x^2}\!\!\cdot\ln 3\right)\,(2x) \;=\;2(\ln4)(\ln3)(x)\left(4^{3^{x^2}}\right)\left(3^{x^2}\right)\)