Question:

Find the second order derivatives of x^2+3x+2

Answer:

Answer: The second order derivative of x^2+3x+2 is 2.

Question:

Find the second order derivatives of tan^−1x

Answer:

Solution:

Step 1: Recall that the derivative of tan^−1x is equal to 1/(1+x^2).

Step 2: Take the derivative of both sides with respect to x, using the chain rule.

Step 3: The derivative of 1/(1+x^2) is equal to -2x/(1+x^2)^2.

Step 4: Therefore, the second order derivative of tan^−1x is equal to -2/(1+x^2)^2.

Question:

Find the second order derivatives of sin(logx)

Answer:

Answer:

1. Differentiate sin(logx) with respect to x: d/dx(sin(logx)) = cos(logx) * d/dx(logx)

2. Differentiate logx with respect to x: d/dx(logx) = 1/x

3. Substitute d/dx(logx) with 1/x in the first step: d/dx(sin(logx)) = cos(logx) * 1/x

4. Differentiate d/dx(sin(logx)) with respect to x: d²/dx²(sin(logx)) = -sin(logx) * (1/x)²

Question:

If y=cos^−1x, find d^2y/dx^2 in terms of y alone.

Answer:

Answer:

d^2y/dx^2 = -sin(y)

Question:

Find the second order derivatives of logx

Answer:

Answer: Step 1: We can use the chain rule to differentiate logx.

Step 2: Let y = logx

Step 3: Differentiate y with respect to x to get the first order derivative:

dy/dx = 1/x

Step 4: Differentiate the first order derivative with respect to x to get the second order derivative:

d2y/dx2 = -1/x2

Question:

Find the second order derivatives of e^xsin5x

Answer:

Answer:

First order derivative: e^xsin5x + 5e^xcos5x

Second order derivative: e^xsin5x + 10e^xcos5x - 25e^xsin5x

Question:

Find the second order derivatives of x^3.logx

Answer:

Answer: Step 1: Differentiate x^3 with respect to x Answer: 3x^2

Step 2: Differentiate logx with respect to x Answer: 1/x

Step 3: Multiply 3x^2 and 1/x Answer: 3x

Step 4: Differentiate 3x with respect to x Answer: 3

Question:

Find the second order derivatives of e^6xcos3x

Answer:

Answer:

Step 1: Differentiate the expression e^6xcos3x with respect to x.

d/dx (e^6xcos3x) = 6e^6xcos3x + e^6x(-3sin3x)

Step 2: Differentiate the expression again with respect to x.

d/dx (6e^6xcos3x + e^6x(-3sin3x)) = 6(6e^6xcos3x) + e^6x(-3cos3x) + (6e^6x(-3sin3x)) + e^6x(-3(-sin3x))

Step 3: Simplify the expression.

d/dx (6e^6xcos3x + e^6x(-3sin3x)) = 36e^6xcos3x - 3e^6xcos3x - 18e^6xsin3x - 3e^6xsin3x

Step 4: Combine the like terms.

d/dx (6e^6xcos3x + e^6x(-3sin3x)) = 33e^6xcos3x - 21e^6xsin3x

Therefore, the second order derivatives of e^6xcos3x is 33e^6xcos3x - 21e^6xsin3x.

Question:

Find the second order derivatives of x.cosx

Answer:

Answer:

1. First order derivative: -x.sinx + cosx
2. Second order derivative: -sinx - x.cosx

Question:

Find the second order derivatives of log(logx)

Answer:

Answer:

Step 1: Use the chain rule to find the first order derivative:

d/dx(log(logx)) = 1/(logx)*(1/x)

Step 2: Use the chain rule again to find the second order derivative:

d2/dx2(log(logx)) = -1/(logx)2*(1/x)2 + 1/(logx)*(-1/x2)

Simplifying:

d2/dx2(log(logx)) = -1/(x2*logx2)

Question:

Find the second order derivatives of x^20

Answer:

1. First order derivative of x^20 = 20x^19

2. Second order derivative of x^20 = (20)(19)(x^18)