Calculate
dy
dx
.
You need not expand your answer.
y = (9x2 + x)(x − x2)
y = (9x^2 + x)(x − x^2)
You could expand the expression to four (reduceable to three) terms and then differentiate term by term, OR use formula for derivative of the product of two functions. The answer will be the same.
They apparently want you to use the second method, which gives
dy/dx = (x - x^2)*(18x + 1) + (9x^2 + x)(1 -2x)
You could expand and combine terms, but they don't ask you to.
Oh, I see you want me to find the derivative of y with respect to x. Okay, let's put on our mathematical clown noses and get started!
Now, the function we're given is y = (9x^2 + x)(x - x^2). To find dy/dx, we'll use the product rule.
So, let's imagine we're multiplying two things - the first one is (9x^2 + x) and the second one is (x - x^2). We'll differentiate each of those separately.
The derivative of (9x^2 + x) with respect to x is 18x + 1. That's like saying if you had a monkey with 9x^2 + x bananas, and it ate all of them, leaving no trace of the bananas, you'd be left with just 18x + 1 monkeys.
Now, the derivative of (x - x^2) with respect to x is simply 1 - 2x. If you had a group of x - x^2 people, and you lined them all up but then some of them backed out, you'd be left with just 1 - 2x people.
Alright, now for the finale! We'll use the product rule. It goes like this:
dy/dx = (first term of the product) times (derivative of the second term)
+ (second term of the product) times (derivative of the first term)
So, applying the product rule, we get:
dy/dx = (9x^2 + x)(1 - 2x) + (x - x^2)(18x + 1)
Now, we've found the derivative of y with respect to x, without expanding our answer. Ta-da!
To find dy/dx, we can use the product rule and chain rule.
Let's start by expanding y = (9x^2 + x)(x - x^2):
y = 9x^3 - 9x^4 + x^2 - x^3
To find dy/dx, we will differentiate each term with respect to x:
d(9x^3)/dx = 27x^2
d(-9x^4)/dx = -36x^3
d(x^2)/dx = 2x
d(-x^3)/dx = -3x^2
Now we can combine these derivatives to find dy/dx:
dy/dx = 27x^2 - 36x^3 + 2x - 3x^2
Simplifying this expression, we get:
dy/dx = -36x^3 + 24x^2 + 2x
To calculate dy/dx, we need to use the product rule of differentiation.
The product rule states that if we have two functions u(x) and v(x), then the derivative of their product, w(x) = u(x) * v(x), is given by:
dw/dx = u(x) * dv/dx + v(x) * du/dx
In this case, y = (9x^2 + x)(x − x^2).
Let's assign u(x) = 9x^2 + x and v(x) = x - x^2.
Now, let's find the derivatives of u(x) and v(x) separately.
To find du/dx, we will differentiate u(x) with respect to x:
du/dx = d/dx (9x^2 + x)
To differentiate 9x^2, we use the power rule, which states that if we have a function f(x) = x^n, then its derivative f'(x) is given by:
f'(x) = n * x^(n-1)
Applying the power rule to 9x^2, we get:
du/dx = d/dx (9x^2) + d/dx (x)
du/dx = 18x^1 + 1
du/dx = 18x + 1
Similarly, to find dv/dx, we will differentiate v(x) with respect to x:
dv/dx = d/dx (x - x^2)
To differentiate x, we get:
dv/dx = d/dx (x) - d/dx (x^2)
dv/dx = 1 - 2x
Now, we can calculate dy/dx using the product rule:
dy/dx = u(x) * dv/dx + v(x) * du/dx
dy/dx = (9x^2 + x)(1 - 2x) + (x - x^2)(18x + 1)
Simplifying this expression would require expanding and combining terms, but since you mentioned not to expand the answer, the expression above is the derivative dy/dx of the function y = (9x^2 + x)(x - x^2).