Find the derivative of f(x) = 2-2^{-x-1}

Question:

Find the derivative of {eq}f(x) = 2-2^{-x-1} {/eq}

Quotient Rule of Derivative:

Consider the functions {eq}f(x) {/eq} and {eq}g(x) {/eq} both are differentiable and {eq}g(x) \neq 0 {/eq} {eq}{\left( {\frac{f}{g}} \right)^\prime } = \frac{{f'\,g - f\,g'}}{{{g^2}}} {/eq}. Use this result for finding the derivative of given function. Use the fact that {eq}d(a^x) = a^x \ln(a) {/eq}

Answer and Explanation:

{eq}\displaystyle f(x) = 2-2^{-x-1} = 2- \frac{1}{2^{x+1}} \\ \displaystyle f'(x) = 0 - \frac{d(2^{x+1})}{(2^{x+1})^2} ... Quotient \ Rule\\ \displaystyle f'(x) = \frac{2 d(2^x)}{(2^{x+1})^2} \\ \displaystyle f'(x) = \frac{2 \cdot 2^x \ln(2)}{4(2^{x})^2} ... \ d(a^x) = a^x \ln(a)\\ \displaystyle f'(x) = \frac{ \ln(2)}{2 \cdot 2^{x}} \\ {/eq}


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When to Use the Quotient Rule for Differentiation

from Math 104: Calculus

Chapter 8 / Lesson 8
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