Question

Differentiate the given function w.r.t. x
$f(x) = e^{ \left (e^{ \left[e^{(4x^2+1)}\right] } \right)}$
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Answer 1

Answer:

$e^{(e^{[e^{(4x^2+1)}]})}$.$e^{[e^{(4x^2+1)}]}$.$e^{(4x^2+1)}$.8x

Step-by-step explanation:

Okay so, We're gonna use chain rule for differentiating this w.r.t. x

This rule is used to differentiate such type of functions that have an inner and an outer function.

I'm gonna try describing this rule so that you use it efficiently on your own to calculate big big problems like this.

Suppose, there's a function

y = f( g(x) )

So, we'll differentiate the outer function that is "f" and then multiply it with the derivative of the inner function {g (x)}.

so we get :-

dy/ dx = f'(g(x)) . g'(x)

[f'(x) is the derivative of f(x)]

Back to the question :-

$y=e^{(e^{[e^{(4x^2+1)}]})}$

• derivative of exponent function is the same as the function.
• derivative of x² is 2x

so outer function is $e^{(e^{[e^{(4x^2+1)}]})}$ whose derivative is also $e^{(e^{[e^{(4x^2+1)}]})}$.

inner function is $e^{[e^{(4x^2+1)}]}$ whose derivative is $e^{[e^{(4x^2+1)}]}$

its inner function is $e^{(4x^2+1)}$ whose derivative is $e^{(4x^2+1)}$

innermost function is 4x² + 1 whose derivative is 4(2x) + 0 = 8x

Now multiplying all these derivatives obtained, as per the chain rule :-

$e^{(e^{[e^{(4x^2+1)}]})}$.$e^{[e^{(4x^2+1)}]}$.$e^{(4x^2+1)}$.8x

Sooo, the derivative of the given function is $e^{(e^{[e^{(4x^2+1)}]})}$.$e^{[e^{(4x^2+1)}]}$.$e^{(4x^2+1)}$.8x.

                                     

Hope this helps! :}