Differentiate.
sqrt((6x^2 + 5x + 1)^3)
Use the chain rule.
Let u(x) = 6x^2 + 5x + 1
f(x) = u(x)^3/2
df/dx = df/du*du/dx
= (3/2)*sqrt(6x^2 + 5x + 1)*(12x + 5)
In google type:
wolfram alpha
When you see list of results click on:
Wolfram Alpha:Computation Knowledge Engine
When page be open in rectangle type:
sqrt((6x^2 + 5x + 1)^3)
and click option =
After few seconds you will see everything about that functions.
Then click option:
Derivative: Shov steps
To differentiate the given expression, we can use two rules of differentiation: the chain rule and the power rule. Here's how you can differentiate the expression step by step:
Step 1: Identify the outermost function and the inner function within it. In this case, the outermost function is the square root (√), and the inner function is (6x^2 + 5x + 1)^3.
Step 2: Apply the chain rule, which states that the derivative of the composite function (f(g(x))) is found by multiplying the derivative of the outer function (f'(g(x))) with the derivative of the inner function (g'(x)). The derivative of a power function (x^n) is given by the power rule, which states that the derivative is equal to the exponent multiplied by the constant coefficient.
Step 3: Find the derivative of the inner function, (6x^2 + 5x + 1)^3. To do this, use the power rule. Multiply the exponent (3) by the coefficient (1) to get 3(6x^2 + 5x + 1)^2.
Step 4: Find the derivative of the outer function, √. To differentiate the square root, divide the derivative of the inner function by twice the square root of the argument. In this case, the argument is (6x^2 + 5x + 1)^3. So, the derivative of √(6x^2 + 5x + 1)^3 is (3(6x^2 + 5x + 1)^2) / (2√(6x^2 + 5x + 1)).
Therefore, the differentiation of sqrt((6x^2 + 5x + 1)^3) is (3(6x^2 + 5x + 1)^2) / (2√(6x^2 + 5x + 1)).