Differentiate y = (sqrt(1+x2)-1x) with respect to x.
|Mathematics
Differentiate y = (sqrt(1+x2)-1x) with respect to x.
ScanSolve AI Answer
Answer
2(1+x2)1
1. Differentiate y=arctan(x1+x2−1) with respect to x.
Step 1: Use a trigonometric substitution to simplify the expression inside the arctan function.
Let x=tanθ.
Then 1+x2=1+tan2θ=sec2θ=∣secθ∣.
For simplicity, assume x>0, which implies θ∈(0,π/2), so secθ>0.
Substitute these into the expression inside the arctan:
x1+x2−1=tanθsecθ−1
Step 2: Simplify the trigonometric expression.
Rewrite secθ and tanθ in terms of sinθ and cosθ:
tanθsecθ−1=cosθsinθcosθ1−1=cosθsinθcosθ1−cosθ=sinθ1−cosθ
Now, use the half-angle identities: 1−cosθ=2sin2(2θ) and sinθ=2sin(2θ)cos(2θ).
sinθ1−cosθ=2sin(2θ)cos(2θ)2sin2(2θ)=cos(2θ)sin(2θ)=tan(2θ)
Step 3: Substitute the simplified expression back into y.
y=arctan(tan(2θ))
Since x=tanθ, for x>0, we have θ=arctanx, and θ∈(0,π/2). This means 2θ∈(0,π/4). In this interval, arctan(tanu)=u.
Therefore,
y=2θ
Step 4: Express y in terms of x.
Since θ=arctanx, substitute this back into the expression for y:
y=21arctanx
This simplification holds for x=0.
Step 5: Differentiate y with respect to x.
dxdy=dxd(21arctanx)
Recall that dxd(arctanx)=1+x21.
dxdy=21⋅1+x21dxdy=2(1+x2)1
The final answer is 2(1+x2)1.
That's 2 down. 3 left today — send the next one.
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This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
1. Differentiate y = (sqrt(1+x^2)-1x) with respect to x. Step 1: Use a trigonometric substitution to simplify the expression inside the function. Let x = . Then sqrt(1+x^2) = sqrt(1+^2) = sqrt(^2) = ||. For simplicity, assume x > 0, which implies (0, /2), so > 0. Substitute these into the expression inside the : sqrt(1+x^2)-1x = ( - 1)/() Step 2: Simplify the trigonometric expression. Rewrite and in terms of and : ( - 1)/() = (1)/() - 1()/() = (1-)/()()/() = (1-)/() Now, use the half-angle identities: 1- = 2^2(()/(2)) and = 2(()/(2))(()/(2)). (1-)/() = (2^2()/(2))2(()/(2))(()/(2)) = (()/(2))(()/(2)) = (()/(2)) Step 3: Substitute the simplified expression back into y. y = ((()/(2))) Since x = , for x > 0, we have = x, and (0, /2). This means ()/(2) (0, /4). In this interval, ( u) = u. Therefore, y = ()/(2) Step 4: Express y in terms of x. Since = x, substitute this back into the expression for y: y = (1)/(2) x This simplification holds for x ≠ 0. Step 5: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)((1)/(2) x) Recall that (d)/(dx)( x) = (1)/(1+x^2). (dy)/(dx) = (1)/(2) · (1)/(1+x^2) (dy)/(dx) = (1)/(2(1+x^2)) The final answer is (1)/(2(1+x^2)). That's 2 down. 3 left today — send the next one.
