9–61. Evaluate and simplify y'.


y = e^sin x+2x+1

Verified step by step guidance

1. First, identify the function y given in the problem. Here, y = e^(sin(x)) + 2x + 1.
2. Next, we need to find the derivative of y with respect to x, denoted as y'. This involves differentiating each term in the function y separately.
3. For the term e^(sin(x)), use the chain rule. The derivative of e^(u) with respect to u is e^(u), and the derivative of sin(x) with respect to x is cos(x). Therefore, the derivative of e^(sin(x)) is e^(sin(x)) * cos(x).
4. For the term 2x, the derivative is straightforward. The derivative of 2x with respect to x is simply 2.
5. The derivative of the constant term 1 is 0, as constants have no rate of change. Combine all these derivatives to express y' as y' = e^(sin(x)) * cos(x) + 2.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the function y = e^(sin x) + 2x + 1 to find y'.

Chain Rule

The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is particularly relevant for the term e^(sin x) in the given function.

Exponential Functions

Exponential functions are functions of the form f(x) = a^x, where a is a constant. In this case, e^(sin x) is an exponential function where the exponent is a function itself (sin x). Understanding how to differentiate exponential functions is crucial for evaluating y' accurately.