Continuity and Differentiability Exercise 07
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:

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

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

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

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:
 First order derivative: x.sinx + cosx
 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:

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

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