Find derivative of
y= x^3(5x-1)^4
I got
y'= 2x(5x-1)^3 +16x^2+5x-1
is this right? If not what and where did I go wrong
for starters
x^3[ 4(5x-1)^3 (5) ] + 3 x^2 (5x-1)^4
Hmmm. You have
y = uv, so
y' = u'v+uv'
= 3x^2(5x-1)^4 + x^3 * 4(5x-1)^3(5)
= x^2(5x-1)^3 (3(5x-1) + 20x)
= x^2(35x-3)(5x-1)^3
To find the derivative of y = x^3(5x-1)^4, you need to use the product rule and the chain rule.
Let's break down the steps:
Step 1: Apply the product rule
The product rule states that if you have a function f(x) = u(x)v(x), where u(x) and v(x) are differentiable functions, then the derivative of f(x) is given by f'(x) = u(x)v'(x) + u'(x)v(x).
For our function y = x^3(5x-1)^4, let's assign u(x) = x^3 and v(x) = (5x-1)^4.
Step 2: Find the derivatives of u(x) and v(x)
Using the power rule, the derivative of u(x) = x^3 is u'(x) = 3x^2.
Using the chain rule, the derivative of v(x) = (5x-1)^4 is v'(x) = 4(5x-1)^3 * 5 = 20(5x-1)^3.
Step 3: Apply the product rule to find y'
Now, we can use the product rule formula to find the derivative of y:
y' = u(x)v'(x) + u'(x)v(x)
= x^3 * 20(5x-1)^3 + 3x^2 * (5x-1)^4
Expanding the above expression yields:
y' = 20x^3(5x-1)^3 + 3x^2(5x-1)^4
So, your derivative expression should be:
y' = 20x^3(5x-1)^3 + 3x^2(5x-1)^4
Therefore, your original expression is incorrect, and you went wrong with the signs and term placement in your derivation.
To find the derivative of the given function, you need to apply the product rule and the chain rule. Let's break down the process step by step:
Step 1: Apply the product rule
The product rule states that if you have two functions u(x) and v(x), the derivative of their product (u(x) * v(x)) is given by:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
In your case, u(x) = x^3 and v(x) = (5x - 1)^4. So, let's differentiate each part:
u'(x) = 3x^2 (using the power rule)
v'(x) = 4(5x - 1)^3 (using the chain rule)
Step 2: Substitute the values back into the product rule formula
Now, let's substitute the values we found back into the product rule formula:
y' = u'(x) * v(x) + u(x) * v'(x)
= 3x^2 * (5x - 1)^4 + x^3 * 4(5x - 1)^3
Step 3: Simplify
To simplify further, we can factor out (5x - 1)^3, since both terms have it:
y' = (5x - 1)^3 * (3x^2 + 4x^3)
= (5x - 1)^3 * (3x^2 + 20x^3)
So, the correct derivative of y = x^3(5x - 1)^4 is y' = (5x - 1)^3 * (3x^2 + 20x^3).
Looking at your answer, it seems you missed applying the power rule to the second term (16x^2) and also didn't factor out (5x - 1)^3 like we did in Step 3. Therefore, your answer is not correct.
I hope this helps clarify the process of finding the derivative and where the mistake occurred.