Differentiate.
sqrt((6x^2 + 5x + 1)^3)
let y = √((6x^2 + 5x + 1)^3)
= (6x^2 + 5x + 1)^(3/2)
dy/dx = (3/2)(6x^2 + 5x + 1)^(1/2)(12x + 5)
= (3/2)(12x+5)√(6x^2 + 5x + 1)
To differentiate the given expression, you can use the chain rule in calculus. The chain rule states that if you have a composite function, the derivative is calculated by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
Let's break down the given expression using the chain rule:
1. Start by identifying the outer function: sqrt(x)
2. Identify the inner function: (6x^2 + 5x + 1)^3
To differentiate the outer function, we need to recall that the derivative of sqrt(x) is (1/2) * x^(-1/2). So, we have:
d(outer function)/dx = (1/2) * (6x^2 + 5x + 1)^(-1/2)
Now, to differentiate the inner function, we need to use the chain rule again:
d(inner function)/dx = 3 * (6x^2 + 5x + 1)^2 * (12x + 5)
Finally, we can apply the chain rule by multiplying the derivatives of the outer and inner functions:
Differentiated expression = d(outer function)/dx * d(inner function)/dx
= (1/2) * (6x^2 + 5x + 1)^(-1/2) * 3 * (6x^2 + 5x + 1)^2 * (12x + 5)
Simplifying the expression further, we get:
Differentiated expression = 3(12x + 5)(6x^2 + 5x + 1)^(3/2) * (6x^2 + 5x + 1)^(-1/2)
Hence, the differentiated expression of sqrt((6x^2 + 5x + 1)^3) is 3(12x + 5)(6x^2 + 5x + 1)^(3/2) * (6x^2 + 5x + 1)^(-1/2).