Differentiate y=x(x4+5)3
d/dx = [x(x4+5)3]
d/dx = [f(x)g(x)] = f(x)g’(x)+g(x)f’(x)
d/dx = [x(x4+5)3] = x • d/dx (x4+5)3 + (x4+5)3 • d/dx x
g’(x) = d/dx (x4+5)3 = 3(x4+5)2 • 4x
f’(x) = d/dx (x) = 1
= (x) • 3(x4+5)2 • 4x + (x4+5)3 • 1 (factor out the common form (x4+5)2 )
= (x4+5)2 [(x)(3)(4x)+ (x4+5)]
=(x4+5)2 (12x2+x4+5) (answer I got)
The right answer =(x4+5)2 (13x4+5)
Just trying to see where I went wrong.
Shouldn't g'= 3(x^4+5)^2 * 4x^3 ?
It is hard to read your work with the exponents not raised or indicated by a ^ symbol. There is a mistake where you take the derivative of (x^4+5)^3. You used the "function of a function" or "chain" rule incorrectly.
Here is what I get:
y = x(x^4 +5)^3
Let f(x) = x and g(x) = (x^4+5)^3
dy/dx = (x^4+5)^3 + 3x(x^4+5)^2*4x^3
= (x^4+5)^2 *[(x^4+5) + 12x^4]
= (x^4+5)^2 *(13x^4+ 5)
8 . 4
- - 2 = -
6 . 3
It seems you made a mistake when calculating the derivative of (x4+5)3. Let's go through the differentiation step again.
To differentiate y = x(x4+5)3, we can use the product rule. According to the product rule, if we have two functions u(x) and v(x), then the derivative of their product is given by:
d/dx (u(x) * v(x)) = u(x) * v'(x) + v(x) * u'(x)
In this case, let's consider u(x) = x and v(x) = (x4+5)3.
First, we need to find the derivative of v(x), which we can denote as v'(x). Using the chain rule, the derivative of (x4+5)3 would be:
v'(x) = 3(x4+5)2 * d/dx (x4+5)
Next, we need to find the derivative of u(x), which is simply 1.
Now, we can substitute u(x), v(x), u'(x), and v'(x) into the product rule formula:
d/dx (x(x4+5)3) = x * [3(x4+5)2 * d/dx (x4+5)] + (x4+5)3 * 1
Now, let's simplify the derivatives:
d/dx (x(x4+5)3) = x * [3(x4+5)2 * 4x] + (x4+5)3
Simplifying further:
= 12x2 * x * (x4+5)2 + (x4+5)3
= 12x3(x4+5)2 + (x4+5)3
= (x4+5)2 * (12x3 + x4+5)
So, the correct answer is (x4+5)2 * (12x3 + x4+5).