Find the derivative of f(x)=(1+7x^2)(x-x^2) by using the product rule and by multiplying first.
multiplying first:
y = x + 7 x^3 - x^2 -7 x^4
so
dy/dx = 1 + 21 x^2 - 2 x - 28 x^3
Product rule
(1+7x^2)(1-2x) + (x-x^2)(14x)
=1 +7 x^2 -2 x -14 x^3 +14 x^2 - 14 x^3
= 1 + 21 x^2 - 2 x -28 x^3 luckily the same
or, just to make things complete, factor and then use the product rule:
f(x) = x(1-x)(1+7x^2)
f'(x) = (1-x)(1+7x^2) + x(-1)(1+7x^2) + x(1-x)(14x)
= 1 - 2x + 21x^2 - 28x^3
To find the derivative of f(x) = (1 + 7x^2)(x - x^2), we can use both the product rule and the method of multiplying first. Let's go through both methods step by step.
Using the product rule:
The product rule states that the derivative of the product of two functions u(x) and v(x) is given by the formula:
(d/dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
In our case, u(x) = (1 + 7x^2) and v(x) = (x - x^2). To find the derivative of f(x), we need to find the derivatives of u(x) and v(x) using the power rule and then apply the product rule.
Step 1: Find the derivative of u(x):
Let's start by finding the derivative of u(x) = (1 + 7x^2).
To do this, we take the derivative of each term separately. The derivative of 1 is 0 because it's a constant, and the derivative of 7x^2 is 14x according to the power rule.
So, u'(x) = 0 + 14x = 14x.
Step 2: Find the derivative of v(x):
Next, let's find the derivative of v(x) = (x - x^2).
Again, we take the derivative of each term separately. The derivative of x is 1, and the derivative of x^2 is 2x according to the power rule.
So, v'(x) = 1 - 2x.
Step 3: Apply the product rule:
Using the product rule formula mentioned earlier, we can now find the derivative of f(x).
f'(x) = u'(x)v(x) + u(x)v'(x)
= (14x)(x - x^2) + (1 + 7x^2)(1 - 2x)
= 14x^2 - 14x^3 + 1 - 2x + 7x^2 - 14x^3
Combining like terms, the final derivative of f(x) is:
f'(x) = 21x^2 - 14x^3 - 2x + 1
Now let's proceed with the method of multiplying first.
Using the method of multiplying first:
This method involves multiplying the two functions together first and then taking the derivative of the resulting function.
Step 1: Multiply the two functions:
f(x) = (1 + 7x^2)(x - x^2)
= x - x^2 + 7x^3 - 7x^4
Step 2: Find the derivative of the product:
Now we can take the derivative of the resulting function.
f'(x) = d/dx[x - x^2 + 7x^3 - 7x^4]
= 1 - 2x + 21x^2 - 28x^3
After simplifying, we obtain the same derivative as before:
f'(x) = 21x^2 - 14x^3 - 2x + 1
Both methods arrive at the same result, confirming the correctness of our derivative.