Explain how to determine derivative of function y=2^(2x+3)?

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shaznl1's profile pic

shaznl1 | High School Teacher | (Level 1) Salutatorian

Posted on

well using the logarithm differentation is right, here is also another way to do it, which is by the exponent rule and the chain rule.

The chain rule states that you split a function into two functions, differentiate the outside function, leave the inside alone, then differentiate the inside function.

2^(2x+3) is basically two functions. One is 2^y , another is y=2X+3

we know by the exponent rule

differentation of A^X (A is a constant)=A^X ln A

so by the chain rule

it is ln 2 * 2^ Y * 2

the two at the end is the dervative of the inside function

well Y= 2x+3

it is 2*ln2* 2^(2X+3)

btw, the log laws state that 2 ln 2 = ln (2^2)=ln 4

so the derivative is

ln 4 *(2^(2x+3))

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, we need to take natural logarithms both sides:

ln y = ln [2^(2x+3)]

We'll use the power property of logarithms:

ln y = (2x+3)*ln 2

We'll differentiate both sides:

y'/y = 2ln 2

y' = (ln4)*y

But y = 2^(2x+3)

y' = (ln4)*[2^(2x+3)]

The requested derivative of the given function is y' = (ln4)*[2^(2x+3)].

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