9–61. Evaluate and simplify y'.


y = e^6x sin x

Verified step by step guidance

1. Step 1: Identify the function y = e^{6x} \(\sin\) x as a product of two functions, u(x) = e^{6x} and v(x) = \(\sin\) x.
2. Step 2: Apply the product rule for differentiation, which states that if y = u(x)v(x), then y' = u'(x)v(x) + u(x)v'(x).
3. Step 3: Differentiate u(x) = e^{6x} with respect to x. Use the chain rule: u'(x) = \(\frac{d}{dx}\)[e^{6x}] = 6e^{6x}.
4. Step 4: Differentiate v(x) = \(\sin\) x with respect to x. The derivative is v'(x) = \(\cos\) x.
5. Step 5: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: y' = 6e^{6x} \(\sin\) x + e^{6x} \(\cos\) x.

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 case, we need to differentiate the function y = e^(6x) sin(x) to find y'. This involves applying rules such as the product rule and the chain rule, which are essential for handling functions that are products of other functions.

Product Rule

The product rule is a formula used to differentiate products of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. In the context of the given function y = e^(6x) sin(x), we will apply the product rule to differentiate the exponential function and the sine function together.

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x. In this problem, the chain rule will be necessary when differentiating the exponential part e^(6x), as it involves an inner function (6x) that also needs to be differentiated.