Calculate the derivative of the following functions.
y = e^2x(2x-7)⁵

Verified step by step guidance

1. Step 1: Identify the structure of the function y = e^{2x}(2x-7)^5. Notice that it is a product of two functions: u(x) = e^{2x} and v(x) = (2x-7)^5.
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^{2x}. Use the chain rule: the derivative of e^{2x} is e^{2x} times the derivative of 2x, which is 2. So, u'(x) = 2e^{2x}.
4. Step 4: Differentiate v(x) = (2x-7)^5. Again, use the chain rule: the derivative is 5(2x-7)^4 times the derivative of the inside function 2x-7, which is 2. So, v'(x) = 10(2x-7)^4.
5. Step 5: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: y' = 2e^{2x}(2x-7)^5 + e^{2x}10(2x-7)^4. Simplify the expression if needed.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. Derivatives are used to find rates of change, optimize functions, and analyze the behavior of functions.

Product Rule

The product rule is a formula used to find the derivative of the product 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'. This rule is essential when differentiating functions that are multiplied together, such as in the given function y = e^(2x)(2x-7)⁵.

Chain Rule

The chain rule is a technique 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 can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is particularly useful for functions raised to a power, like (2x-7)⁵ in the given problem.