Question

Differentiate: x^2(1-2x)^1/3​

Answer 1

Answer:

How do I differentiate x2(2x+1)3 ?

Short answer:

2x(2x+1)2(5x+1)

The fairy tale:

First and foremost, you have to identify the form of the function!

Is it a product of two terms, a function raised to a power or in the term of a quotient?

Being the first case, we apply its rule:

Let v=(2x+1)3

Rule : (uv)′=u′v+v′u

So we differeniate each of u and v separetely since we need it in the rule

Again, we identify the form each of u and v before differeniating them

Both of them are functions raised to powers

So the rule goes as such: um=mu(m−1)u′ where :

u is a function (x in case of u; (2x+1) in case of v)

m is the power of the function (2 in case of u, 3 in case of v)

u’ is the derivative of the function u (1 in case of x, 2 in case of (2x+1)

So we get:

u=x2

u′=2x2−11=2x

And v=(2x+1)3

v′=3(2x+1)3−12=6(2x+1)2

Finally, uv=u′v+v′u=2x(2x+1)3+6x2(2x+1)2

Taking ( 2x+1)2 as a common factor

We'll get, (2x+1)2(4x2+2x+6x2)=(2x+1)2(2x+10x2)

For more simplification, we can take 2x common from the second factor,

Final answer:

2x(2x+1)2(5x+1)