product rule examples
Sure! Here are a few examples of how to use the product rule:
Example 1:
Find the derivative of f(x) = x^3 * sin(x).
Solution:
Let u = x^3 and v = sin(x).
Using the product rule: f'(x) = u'v + uv'
Next, find the derivatives of u and v:
u' = 3x^2 (the power rule for differentiation)
v' = cos(x) (the derivative of sin(x) is cos(x))
Now substitute these values into the product rule:
f'(x) = (3x^2)(sin(x)) + (x^3)(cos(x))
So the derivative of f(x) is: f'(x) = 3x^2sin(x) + x^3cos(x).
Example 2:
Find the derivative of f(x) = (2x^2 + 3)(4x - 5).
Solution:
Let u = 2x^2 + 3 and v = 4x - 5.
Using the product rule: f'(x) = u'v + uv'
Next, find the derivatives of u and v:
u' = 4x (the power rule for differentiation)
v' = 4 (the derivative of 4x is 4)
Now substitute these values into the product rule:
f'(x) = (4x)(4x - 5) + (2x^2 + 3)(4)
Simplifying, we get: f'(x) = 8x^2 - 20x + 8x^2 + 12
Combining like terms, we get: f'(x) = 16x^2 - 20x + 12.
These are just a couple of examples of how to use the product rule. Just remember to identify the two functions (u and v), find their derivatives (u' and v'), and apply the product rule formula: f'(x) = u'v + uv'.
with just numbers
Certainly! Here are a couple of examples using numbers only:
Example 1:
Find the derivative of f(x) = 2x * 3.
Solution:
Let u = 2x and v = 3.
Using the product rule: f'(x) = u'v + uv'
Next, find the derivatives of u and v:
u' = 2 (the derivative of 2x is 2)
v' = 0 (the derivative of a constant is always zero)
Now substitute these values into the product rule:
f'(x) = (2)(3) + (2x)(0)
Simplifying, we get: f'(x) = 6 + 0
So the derivative of f(x) is: f'(x) = 6.
Example 2:
Find the derivative of f(x) = (4x - 3)(2x + 5).
Solution:
Let u = 4x - 3 and v = 2x + 5.
Using the product rule: f'(x) = u'v + uv'
Next, find the derivatives of u and v:
u' = 4 (the derivative of 4x is 4)
v' = 2 (the derivative of 2x is 2)
Now substitute these values into the product rule:
f'(x) = (4)(2x + 5) + (4x - 3)(2)
Simplifying, we get: f'(x) = 8x + 20 + 8x - 6
Combining like terms, we get: f'(x) = 16x + 14.
These examples show how to apply the product rule when dealing with simple numerical expressions.
Sure! Here are a few examples of using the product rule in calculus:
Example 1:
Find the derivative of f(x) = x^2 * sin(x).
Step 1: Identify the two functions being multiplied together: f(x) = x^2 and g(x) = sin(x).
Step 2: Differentiate the first function f'(x) = 2x.
Step 3: Keep the second function as it is: g(x) = sin(x).
Step 4: Apply the product rule formula: (f * g)' = f' * g + f * g', where ' denotes differentiation.
Step 5: Substitute the values into the formula: f'(x) * g(x) + f(x) * g'(x) = (2x * sin(x)) + (x^2 * cos(x)).
Therefore, the derivative of f(x) = x^2 * sin(x) is f'(x) = 2x * sin(x) + x^2 * cos(x).
Example 2:
Find the derivative of f(x) = x^3 * e^x.
Step 1: Identify the two functions being multiplied together: f(x) = x^3 and g(x) = e^x.
Step 2: Differentiate the first function f'(x) = 3x^2.
Step 3: Keep the second function as it is: g(x) = e^x.
Step 4: Apply the product rule formula: (f * g)' = f' * g + f * g', where ' denotes differentiation.
Step 5: Substitute the values into the formula: f'(x) * g(x) + f(x) * g'(x) = (3x^2 * e^x) + (x^3 * e^x).
Therefore, the derivative of f(x) = x^3 * e^x is f'(x) = 3x^2 * e^x + x^3 * e^x.
I hope these examples help illustrate how to use the product rule! Let me know if you have any further questions.
The product rule is a rule of differentiation used in calculus to find the derivative of a product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function.
To better understand the product rule, let's go through a couple of examples:
Example 1:
Let's find the derivative of the function f(x) = x^2 * sin(x).
Step 1: Identify the two functions being multiplied together: f(x) = x^2 and g(x) = sin(x).
Step 2: Find the derivatives of both functions: f'(x) = 2x and g'(x) = cos(x).
Step 3: Apply the product rule formula: (f * g)' = f' * g + f * g'
= (x^2 * sin(x))' = (2x * sin(x)) + (x^2 * cos(x))
= 2x * sin(x) + x^2 * cos(x)
So, the derivative of f(x) = x^2 * sin(x) is f'(x) = 2x * sin(x) + x^2 * cos(x).
Example 2:
Let's find the derivative of the function h(x) = (2x^3 + 5x) * e^x.
Step 1: Identify the two functions being multiplied together: h(x) = (2x^3 + 5x) and g(x) = e^x.
Step 2: Find the derivatives of both functions: h'(x) = 6x^2 + 5 and g'(x) = e^x.
Step 3: Apply the product rule formula: (h * g)' = h' * g + h * g'
= ((2x^3 + 5x) * e^x)' = (6x^2 + 5) * e^x + (2x^3 + 5x) * e^x
= (6x^2 + 5 + 2x^3 + 5x) * e^x
= (2x^3 + 6x^2 + 5x + 5) * e^x
So, the derivative of h(x) = (2x^3 + 5x) * e^x is h'(x) = (2x^3 + 6x^2 + 5x + 5) * e^x.
By following these steps, you can apply the product rule to find the derivative of any product of two functions.