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glasskerfu

Answered question

2021-10-19

Find the derivatives of the following function.

y = 4^{3 \cdot x^{2}}

y'=

sweererlirumeX

Skilled2021-10-20Added 91 answers

Step 1
To find:
The derivative of

y = 4^{3 \cdot x^{2}}

Formula used:
Chain rule of derivative:

{(f (g (x)))}^{'} = f^{'} (g (x)) \cdot g (x)

The derivative of

a^{x} i s a^{x} \cdot \ln a

.
Power rule of derivative:

{\ln a}^{b} = b \cdot \ln a

Step 2
Calculation:
The function

y = 4^{3 x^{2}}

can be written as,

y = 64^{x^{2}}

The derivative of

y = 64^{x^{2}}

can be obtained as,

y^{'} = 64^{x^{2}} \cdot \ln 64 \cdot \frac{d}{d x} (x^{2})

= 2 \cdot x \cdot 64 x^{2} \cdot {\ln 2}^{6}

= 12 x \cdot 64^{x^{2}} \cdot \ln 2

Thus, the derivative of given function is

y^{'} = 12 x \cdot 64^{x^{2}} \cdot \ln 2

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