Find dy/dx if y= x^6[(x^2-9)^8]?
So the answer is supposed to be 2x^5[(x^2-9)^7](11x^2-27), but I don't understand how to get it. Do you use product rule, chain rule or both?
thanks
Actually I figured out that you use both, and ended up with 16x^7(x^2-9)^7 + 6x^5(x^2-9)^8.
but I don't understand how that simplifies to 2x^5(x^2-9)^7(11x^2-27)?
factor out 2x^5(x^2-9)^7 and you have
2x^5(x^2-9)^7 * (8x^2 + 3(x^2-9))
= 2x^5(x^2-9)^7 (8x^2+3x^2-27)
and voila...
To find the derivative of y = x^6[(x^2-9)^8], we need to use both the product rule and the chain rule.
Let's break down the function first:
y = x^6[(x^2-9)^8]
The function y is a product of two functions: f(x) = x^6 and g(x) = (x^2-9)^8.
To find the derivative of a product of two functions, we use the product rule, which states:
(d/dx)(f(x)g(x)) = f(x)(d/dx)(g(x)) + g(x)(d/dx)(f(x))
Now, let's find the derivative step by step.
First, we differentiate the function f(x) = x^6 using the power rule:
(d/dx)(x^6) = 6x^(6-1) = 6x^5
Next, we differentiate the second function g(x) = (x^2-9)^8 using the chain rule. The chain rule states:
(d/dx)(f(g(x))) = f'(g(x)) * g'(x)
Let's separate g(x) into two parts: u = x^2-9 and v = u^8.
First, we find the derivative of u with respect to x:
(d/dx)(x^2-9) = 2x
Next, we find the derivative of v with respect to u:
(d/du)(u^8) = 8u^(8-1) = 8u^7
Now, we can find the derivative of g(x) using the chain rule:
(d/dx)((x^2-9)^8) = 8(x^2-9)^7 * 2x
Finally, we apply the product rule to find the derivative of y:
(d/dx)(y) = f(x)(d/dx)(g(x)) + g(x)(d/dx)(f(x))
= x^6 * (8(x^2-9)^7 * 2x) + (x^2-9)^8 * 6x^5
= 2x^7(x^2-9)^7 + 6x^5(x^2-9)^8
Simplifying this expression further will give us the final answer:
dy/dx = 2x^7(x^2-9)^7 + 6x^5(x^2-9)^8
Therefore, the derivative of y with respect to x is 2x^7(x^2-9)^7 + 6x^5(x^2-9)^8.