Can anyone plz explain how to find the derivative using the chain rule for this function?

Question

y= e^4tan√x -- plz explain clearly at the chain rule part b/c that's the part i dont really get. Thnx!

Answer 1

once you know the chain rule , the ans is quite simple. first, to
take derivative of a function, identify , the constants-- in your case e^4
thus, this will come out of the derivative
d/dx(y)=e^4d/dx(tan(x)^(1/2))
now to apply chain rule , let x^(1/2)=u
chain rule staes that
d/dx(y)=d/du(f(u)) X d/dx(f(x)) .............f(x)is function of x

thus,
d/dx(tan (sqrt(x))=d/du(tan(u)) X d/dx(sqrt(x))
...............=(sec(u))^2 X (1/2)x^(-1/2)

(TAKING THE DERIVATIVE)
now, subs value of u,
d/dx(tan(sqrt(x))=(sec(sqrt(x) ) ^2 X ((1/2)x^(-1/2)

subs this derivative in the eqn

d/dx(y)=e^4.{[sec(sqrt(x)]^2 X [1/2]x^(-1/2)

Answer 2

y = e^(4 tan(√x))

The chain rule goes as follows: whenever you're differentiating a function expressed in anything other than x, you differentiate normally, BUT take the derivative of the insides.

Example #1:
We know that when y = sin(x), then dy/dx = cos(x). What if we had something like y = sin(x^2)? Then:

y = sin(x^2). dy/dx = cos(x^2) ... {but it's not x, so "chain" by taking the derivative of x^2}.
dy/dx = cos(x^2) (2x), which is the same as
dy/dx = 2x cos(x^2)

Example #2: y = e^[3x^2 + 8x]
The derivative of e^x is e^x; use this fact, and then chain it; multiply by the derivative of the power itself.

dy/dx = e^[3x^2 + 8x] [ 6x + 8]. Simplified,
dy/dx = [6x + 8] e^(3x^2 + 8x)

Back to your original question:

y = e^(4 tan(√x))

We require three derivatives:
1) d/dx e^x = e^x
2) d/dx tan(x) = sec^2(x)
3) d/dx √x = 1/[2√x]

We also use the knowledge that a constant tacked onto a function is ignorable when differentiating. Our answer is

dy/dx = e^(4 tan(√x)) [4 sec^2(√x)] [ 1/[2√x] ]

Simplified, we should get

dy/dx = [ 2 sec^2(√x) e^(4 tan(√x)) ] / √x

Answer 3

fairly conversing, the chain rule may well be state simply by fact the spinoff of a complicated function is the "spinoff of the exterior circumstances the spinoff of the interior. the 1st project i love to do with any function that has stuff interior the denominator is to convey it to the numerator place: f(x) = a million / ( (x^5 + 3) ^ 4) = (x^5 +3)^ (-4) (be conscious the -4 rather of four) So to your function the interior function is g = (x^5 +3) and the exterior function is g^-4. Taking the spinoff then turns into df/dx = (-4)*(x^5+3)^(-5) * (5x^4) simplifying df/dx = -20x^4 / ( (x^5+3) ^ 5)

Answer 4

y= e^4tan√x

f(x)= e^x_____________f.(x)= e^x
g(x)= 4x _____________g.(x)= 4
h(x)= tanx____________h.(x)= (secx)^2
k(x)= √x_____________k.(x)= (1/(2√x))

Note: f. means f prime

Chain Rule: If h(x)=(f(g(x)), then h.(x)=f.(g(x))•g.(x)

Ok, so time to put this rule into action!

y.= (e^4tan√x) • (4)• (sec√x)^2 • (1/(2√x))

Pretty sure this is right…put the answer in any order you want.

I hope you understand it now!

Answer 5

y = e^(4tan√x)
Let u = 4 tan√x
y = e^u
dy / du = e^(u) = e^(4 tan√x)
Let w = √x
dw/dx = (1/2).x^(- 1/2) = 1 / (2√x)
u = 4 tan w
du / dw = 4 sec² w
du/dx = (du/dw).(dw/dx)
du/dx = 4 sec²√x / (2√x) = 2 sec²√x / √x
dy / dx = (dy / du).(du / dx)
dy / dx = e^(4 tan√x) . (2 sec²√x) / √x