Proof of differential formula of inverse trigonometric function Complete steps to prove d (arctan x) / DX = 1 / (1 + x ^ 2) The two formulas cos ^ 2 y + sin ^ 2 y = 1 and 1 + Tan ^ 2 y = sec ^ 2 y can be used

Proof of differential formula of inverse trigonometric function Complete steps to prove d (arctan x) / DX = 1 / (1 + x ^ 2) The two formulas cos ^ 2 y + sin ^ 2 y = 1 and 1 + Tan ^ 2 y = sec ^ 2 y can be used

prove:
Arctan is strictly monotonic and differentiable on R, and Tan x is monotonic and differentiable on (- π / 2, π / 2). There are:
arctan'x=1/(tan'y)=1/sec^2(y)=cos^2(y)
Cos'y = - 1 / root sign (1-y ^ 2)
So arctan'x = 1 / (1 + x ^ 2)
Is there anything you don't understand? I added

Inverse trigonometric function limit X tends to 0 to find Lim X / arc2x limit

Now arcsin2x = t, 2x = Sint, x = 1 / 2sint,
When x tends to 0, t tends to 0
X tends to 0 for LIM X / arc2x limit = t tends to 0 for LIM 1 / 2sint / t limit = 1 / 2

The train shall whistle before entering the tunnel. The driver whistles 262.5m away from the tunnel entrance. After 1.5s, he hears the sound reflected by the cliff at the tunnel entrance. What is the speed of the train?

Suppose the train running speed is V1, the distance between the train and the cliff when the train whistles, s, ∵ v = st, ∵ the distance traveled by the train within 1.5s: S1 = v1t, the total distance traveled by the whistle within 1.5s: S2 = v2t, according to the meaning of the question: S1 + S2 = 2S, i.e. v1t + v2t = 2S, v1 × 1.5s+340m/s × 1.5s=2 × 262.5m, the solution is:

A train drove to the tunnel entrance under the hanging wall at the speed of 20 meters per second. When it was 720 meters away from the hanging wall, the train whistled. How many seconds did the driver hear the echo? (the sound speed is 340 meters per second) find the specific process and analysis

The total distance traveled by train and sound wave is 720 * 2 = 1440m
20t+340t=1440
The solution is t = 4S

At a crossing, it took another 48 seconds after people listened to the train whistle, and the car passed the crossing. The sound speed was 340 meters per second. When the car whistle, it was 1020 meters away from the crossing. How many meters per hour?

The drawing is easy to understand. Draw a straight line. The right side is the crossing, the left side is the starting point of the train, and the distance between them is 1020 meters. If you want to know the speed of the train, just find out the time spent by the train on this road. The whistle goes to people's ears, that is, the time required for the whistle to come 1020 meters from the crossing is 1020 ÷ 340 = 3. That

A train is moving forward at the speed of 20m / s, and the driver whistles 500m away from the tunnel entrance. When an echo is heard, how far is the locomotive from the tunnel?

Suppose that when an echo is heard, the distance between the locomotive and the tunnel is s, the distance the train passes is 500m-s, which is obtained from v = st, and the time taken is T1 = s1v1 = 500m − s20m / S; The distance of sound passing is s + 500m, and the time taken is T2 = s2v2 = 500m + s340m / s. according to the meaning of the topic, the sound is equal to the time taken by the train, then: 500