What's the derivative of y=400(2)^{x+3}?

Question:

What's the derivative of {eq}y=400(2)^{x+3} {/eq}?

Differentiation:

Constant to the power function :

For solving such questions firstly we need to take log of both sides. After taking log we will find power function is no more in power and so complexity is removed.

Further differentiate normally as we do.

Answer and Explanation:

We have,

{eq}y=400(2)^{x+3} {/eq}

now,

take log both sides,

{eq}log y= log 400 + (x+3)log 2 \\ log y= log (2^2 \times 10^10) + (x+3)log 2 \\ log y= 2log 2 + 10 log 10 + (x+3)log 2 \\ {/eq}

Now,

Differentiating both sides,

{eq}\displaystyle \frac{y'}{y} = (x+3)' log 2 \\ \displaystyle \frac{y'}{y} = log 2 \\ \displaystyle y' = ylog 2 \\ \displaystyle y' = (400(2)^{x+3})log 2 \\ {/eq}

so,

{eq}\therefore \color{blue}{\displaystyle y' = 277.25(2)^{x+3} } {/eq}


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Basic Calculus: Rules & Formulas

from Calculus: Tutoring Solution

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