Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the derivative of $$y = 2^{3^{x^{2}}}$$ using the chain rule?

Please show step by step solution.

share|improve this question
2  
what steps did you attempt on your own? –  Sasha Nov 6 '12 at 18:16
    
I'd start with writing the function in the base e, and then go on. –  TheClock Nov 6 '12 at 18:19
add comment

2 Answers

up vote 1 down vote accepted

$$y = 2^{3^{x^{2}}}$$

$$y' = 2^{3^{x^{2}}}\ln2 (3^{x^{2}})'$$

$$y' = 2^{3^{x^{2}}}\ln2 (3^{x^{2}}\ln3)(x^2)'$$

$$y' = 2^{3^{x^{2}}}\ln2 (3^{x^{2}}\ln3)2x$$

$$y' = 2x\ln2 \ln3 2^{3^{x^{2}}}(3^{x^{2}})$$

$$y' = 2^{3^{x^{2}}}3^{x^{2}}2x\ln3\ln2 $$

share|improve this answer
    
Why didn't you leave in the (x^2) between step 3 and 4 as you did with the (3^(x^2)) between steps 2 and 3? –  user44816 Nov 6 '12 at 18:34
    
@user44816 He used the chain rule again, so he differentiated $(3^{x^2})$ into $((3^{x^2}\ln 3) (x^2)')$, and then at the next step he differentiated the last $x^2$. –  Arthur Nov 6 '12 at 18:46
    
If you are leaving in the (3^(x^2)) when differentiating (derivative is (ln3)(x^2), why wouldn't you leave in the (x^2) when differentiating at the next step (becomes 2x)? What is the difference? –  user44816 Nov 6 '12 at 18:52
add comment

Hint: for a function $a^u$, $$\frac{d}{du} = a^u \ln a \cdot u'$$

In your case, $a=2, u = 3^{x^2}$.

share|improve this answer
    
Is $u=3^{x^2}$ not $u=3x^2$ –  Adi Dani Nov 6 '12 at 18:37
    
Yup, typo. Thanks. –  Joe Nov 6 '12 at 18:40
    
Shouldn't it be $\frac{d}{dx}$? If the derivative were with respect to $u$ then $u'=1$ and there's no point writing it. –  Daniel Littlewood Nov 6 '12 at 18:45
    
I replaced his function $3^{x^2}$ as $u$. If I were to write $\frac{d}{dx} a^{u}$, it would not make sense since you are not differentiating with respect to $u$. Also, consider the example with $u = x.$ Then $\frac{d}{du} = 2^x \ln 2$. –  Joe Nov 6 '12 at 18:47
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.