Differentiation questions.?

Question

Help most welcome. Differentiate step by step:

1) arctan(2x)
2) e^(x^2+2)
3) arctan(2x)e^(x^2+2)

Answer 1

Second one, split it to be e^2 * e^(x^2). The derivative of e^2 is just some constant., so it falls out.

The derivative of e^x is e^x * dx/dt a
and the derivative of of x^2 is 2x, so it becomes

e^(X^2) * 2x.

The third one is going to be the product rule, where you take the first one * derivative of the second, plus the Second * derivative of the first.

The other guy gave the derivative of the first one, so you should just be able to plug that in using the product rule.

Answer 2

1) arctan(2x)
Let u = 2x
Then u = tan y and du/dy = sec^2y
dy/du = 1/dU/dy = 1/sec^2y = 1/(1+tan^2y) = 1/(1+u^2)
Hence dy/dx arctan u = 1/(1+u^2) * du/dx [ u is a f(x)=2x]
So just substitute 2x for u and you have your answer.

2) e^(x^2+2)
dy/dx = 2xe^(x^2+2)

3) arctan(2x)e^(x^2+2)
Just use the product rule since you now know the derivatives of arctan(2x) and e^(x^2+2).

Answer 3

First one
y = arctan(2x)
2x = tan(y)
2 = sec^2 y * dy/dx
= (1 + tan^2 y) * dy/dx
= (1 + 4x^2) * dy/dx

dy/dx = 2 / (1 + 4x^2)