If y = (((3x^2)+ 5)^5)((x+2)^4),
then dy/dx = ?
I've gotten to the point where i have
(5((3x^2)+ 5)^4)(4(x+2)^3)(6x), what do i do next? It's a multiple choice question anf this is not one of the answers listed. Any help would be appreciated. Thanks!
Your "first-line" derivative is
dy/dx = 5(3x^2 + 5)^4(6x)(x+2)^4 + 4(x+2)^3(3x^2+5)^5
= 2(x+2)^3(3x^2 + 5)^4 [ 15x(x+2) + 2(3x^2+5) ]
= 2(x+2)^3(3x^2 + 5)^4 [ 15x^2 + 30x + 6x^2 + 10 ]
= 2(x+2)^3(3x^2 + 5)^4 (21x^2 + 30x + 10)
Thank you!
To find the derivative of the given expression y = (((3x^2) + 5)^5)((x + 2)^4), you need to apply the chain rule.
The chain rule states that if u = f(g(x)), then du/dx = f'(g(x)) * g'(x).
First, let's identify the two functions within our expression:
f(u) = u^5
g(x) = (3x^2) + 5
Applying the chain rule, we get:
dy/dx = f'(g(x)) * g'(x)
To find f'(u), we differentiate f(u) with respect to u:
f'(u) = 5u^4
To find g'(x), we differentiate g(x) with respect to x:
g'(x) = 6x
Now, substitute f'(g(x)) = 5((3x^2) + 5)^4 and g'(x) = 6x into our expression:
dy/dx = 5((3x^2) + 5)^4 * 6x
Simplifying further, we get:
dy/dx = 30x((3x^2) + 5)^4
So, the correct answer is dy/dx = 30x((3x^2) + 5)^4.