Under differential calculus using the product rule find the derivative of (2x+3)^3(4x^2-1)^2
first line derivative:
(2x+3)^3(2)(4x^2 - 1)(8x) + (4x^2 - 1)^2(3)(2x+3)^2(2)
take out some common factors and simplify
To find the derivative of the given expression using the product rule in differential calculus, follow these steps:
Step 1: Identify the two functions that are being multiplied together.
In this case, the two functions are (2x+3)^3 and (4x^2-1)^2.
Step 2: Apply the product rule.
The product rule states that if you have two functions, u(x) and v(x), the derivative of their product, (u * v), is given by the formula:
(uv)' = u'v + uv'
where u' and v' represent the derivatives of u(x) and v(x) respectively.
Step 3: Find the derivatives of the two functions.
To calculate the derivatives, we can use the power rule and the chain rule. Let's first find the derivative of (2x+3)^3 and (4x^2-1)^2:
For (2x+3)^3:
Using the power rule, the derivative of (2x+3)^3 is 3(2x+3)^2 times the derivative of (2x+3).
Derivative of (2x+3) = 2.
So, the derivative of (2x+3)^3 is 3(2x+3)^2 * 2.
For (4x^2-1)^2:
Using the power rule, the derivative of (4x^2-1)^2 is 2(4x^2-1) times the derivative of (4x^2-1).
Derivative of (4x^2-1) = 8x.
So, the derivative of (4x^2-1)^2 is 2(4x^2-1) * 8x.
Step 4: Apply the formula for the product rule.
Using the product rule formula (uv)' = u'v + uv', where u = (2x+3)^3 and v = (4x^2-1)^2, and substituting the derivatives we found in Step 3, we can calculate the derivative of the given expression:
[(2x+3)^3(4x^2-1)^2]' = (3(2x+3)^2 * 2)(4x^2-1)^2 + (2x+3)^3 * 2(4x^2-1) * 8x
Simplifying this expression will give you the final answer, which is the derivative of (2x+3)^3(4x^2-1)^2.