Answer:
To differentiate the function y = √x(1 + 2x)², we can use the product rule and chain rule of differentiation.
Step 1: Apply the product rule.
The product rule states that if we have a function u(x) multiplied by another function v(x), the derivative of the product is given by:
d/dx(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
In this case, u(x) = √x and v(x) = (1 + 2x)².
Step 2: Find the derivatives of u(x) and v(x).
The derivative of √x can be found using the power rule of differentiation:
d/dx(√x) = (1/2)x^(-1/2)
The derivative of (1 + 2x)² can be found using the chain rule:
d/dx((1 + 2x)²) = 2(1 + 2x) * (d/dx(1 + 2x)) = 2(1 + 2x) * 2 = 4(1 + 2x)
Step 3: Apply the product rule formula.
Using the product rule formula, we can differentiate y = √x(1 + 2x)² as follows:
dy/dx = (1/2)x^(-1/2) * (1 + 2x)² + √x * 4(1 + 2x)
Step 4: Simplify the expression.
To simplify the expression further, we can expand the square term (1 + 2x)² and combine like terms:
dy/dx = (1/2)x^(-1/2) * (1 + 4x + 4x²) + 4√x(1 + 2x)
dy/dx = (1/2)x^(-1/2) + 2x^(1/2) + 2x^(3/2) + 4√x + 8x√x
Step 5: Arrange the terms in a simplified form.
To further simplify, we can arrange the terms in a simplified form:
dy/dx = 2x^(-1/2) + 2x^(1/2) + 2x^(3/2) + 4x^(1/2) + 8x√x
Finally, we have the derivative of the function y = √x(1 + 2x)² as:
dy/dx = 2x^(-1/2) + 2x^(1/2) + 2x^(3/2) + 4x^(1/2) + 8x√x