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Calculate $\frac{d^{2} y}{d x^{2}}$. $y=e^{-(x-1)}-x$

   Calculate $\frac{d^{2} y}{d x^{2}}$.
$y=e^{-(x-1)}-x$

Finite Mathematics and Applied Calculus

Finite Mathematics and Applied Calculus

Stefan Waner, Steven… 5th Edition

Chapter 12, Problem 7

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Solved by Expert Gregory Higby

09/28/2021

Step 1

We can use the chain rule for the first term and the power rule for the second term. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. The power rule states that Show more…

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Calculate $\frac{d^{2} y}{d x^{2}}$. $y=e^{-(x-1)}-x$

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Key Concepts

Exponential Functions

Exponential functions are functions in which a constant base, often the natural number e, is raised to a variable exponent. Differentiating these functions typically involves applying the chain rule when the exponent is a function of the variable, making them an essential topic in calculus.

Chain Rule

The chain rule is a fundamental technique used for differentiating composite functions. When a function is expressed as a composition of two or more functions, the chain rule allows one to differentiate by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

Differentiation of Linear Functions

Linear functions, which are of the form f(x) = mx + b, have a constant rate of change. Their derivative is straightforward to compute as it equals the slope m, reflecting the constant rate of change of the function.

Differentiation

Differentiation is the process of calculating the derivative of a function, which indicates how the function’s output changes with respect to changes in its input. It is a core concept in calculus used to analyze dynamic changes and rates.

Second Derivative

The second derivative is found by differentiating the first derivative of a function. It measures the rate at which the first derivative changes, providing important information about the curvature of the function and its concavity or convexity.

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