product rule problem example
Here's an example:
Let's say we have the function f(x) = (3x^2 + 2x)(4x^3 - 5x).
To differentiate this function using the product rule, we follow these steps:
1. Identify the two functions being multiplied together: (3x^2 + 2x) and (4x^3 - 5x).
2. Differentiate the first function with respect to x:
- The derivative of 3x^2 + 2x with respect to x is 6x + 2.
3. Keep the second function as it is.
4. Differentiate the second function with respect to x:
- The derivative of 4x^3 - 5x with respect to x is 12x^2 - 5.
5. Apply the product rule formula:
- The derivative of the product of two functions is given by (first function derivative)(second function) + (first function)(second function derivative).
So, applying the product rule to the given function, we have:
f'(x) = (6x + 2)(4x^3 - 5x) + (3x^2 + 2x)(12x^2 - 5).
Simplifying further if needed, but this is the general form of the derivative using the product rule for the given function.
Sure! Here's an example problem that uses the product rule:
Find the derivative of the function f(x) = (2x^3 - 3x) * (4x^2 + 5x).
To find the derivative of a product, we can use the product rule, which states:
If u(x) and v(x) are two functions of x, then the derivative of their product u(x) * v(x) is given by:
(d/dx) [ u(x) * v(x) ] = u'(x) * v(x) + u(x) * v'(x)
Let's find the derivative step by step using the product rule:
Step 1:
Let u(x) = 2x^3 - 3x
Let v(x) = 4x^2 + 5x
Step 2:
Find the derivatives of u(x) and v(x):
u'(x) = 6x^2 - 3 (by applying the power rule and constant rule)
v'(x) = 8x + 5 (by applying the power rule and constant rule)
Step 3:
Now, we can substitute the values into the product rule formula:
(d/dx) [ u(x) * v(x) ] = (6x^2 - 3) * (4x^2 + 5x) + (2x^3 - 3x) * (8x + 5)
Step 4:
Simplify the expression by multiplying and combining like terms:
(d/dx) [ u(x) * v(x) ] = (24x^4 + 30x^3 - 12x^2 - 15x) + (16x^4 + 20x^3 - 24x^2 - 15x)
Step 5:
Combine like terms once again:
(d/dx) [ u(x) * v(x) ] = 40x^4 + 50x^3 - 36x^2 - 30x
So, the derivative of f(x) = (2x^3 - 3x) * (4x^2 + 5x) is f'(x) = 40x^4 + 50x^3 - 36x^2 - 30x.
Sure! I can provide you with an example problem that involves using the product rule.
Example problem: Find the derivative of the function f(x) = (3x^2 + 2x)(4x - 5).
To solve this problem using the product rule, follow these steps:
1. Identify the two functions that are being multiplied together: f(x) = (3x^2 + 2x) and (4x - 5).
2. Apply the product rule formula, which states that the derivative of the product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.
3. Differentiate each function separately.
First, let's find the derivative of the function (3x^2 + 2x):
- The derivative of 3x^2 is 6x (since the power rule states that the derivative of x^n is nx^(n-1)).
- The derivative of 2x is 2 (since the derivative of a constant times x is the constant).
Now, let's find the derivative of the function (4x - 5):
- The derivative of 4x is 4 (since the derivative of a constant times x is the constant).
- The derivative of -5 is 0 (since -5 is a constant, so its derivative is zero).
4. Now, we can apply the product rule to find the derivative of the entire function:
- (3x^2 + 2x)(4x - 5) can be written as (3x^2 + 2x) * 4x - (3x^2 + 2x) * 5
- Using the product rule formula, the derivative is (6x * (4x - 5)) + ((3x^2 + 2x) * 4) - simplified as 24x^2 - 30x + 12x^2 + 8x - 15x^2 - 10x = 21x^2 - 30x + 8x.
Therefore, the derivative of f(x) = (3x^2 + 2x)(4x - 5) is 21x^2 - 30x + 8x.
I hope this example helps you understand how to use the product rule in solving derivative problems!