Differentiation

Step 1: Differentiate .
We can rewrite . Differentiating with respect to , we get:

Step 2: Conclusion.
Thus, .

Step 1: Understand the function.
The function is . For a function to be increasing, its derivative must be positive.

Step 2: Differentiate the function.
The derivative of with respect to is:

Step 3: Analyze the derivative.
Since is always positive for all values of , and the constant factor is also positive, we have:

Step 4: Conclusion.
Thus, the function is increasing on .

Step 1: Applying the chain rule.
We are given the function . To differentiate this with respect to , we use the chain rule. Let , so that . Now, differentiate with respect to and then with respect to : and

Step 2: Differentiating with respect to .
Using the chain rule:

Step 3: Conclusion.
Thus, the derivative of with respect to is .

Step 1: Differentiation of inverse sine.
We have . To differentiate this, we will use the chain rule and the derivative formula for , which is: where .

Step 2: Differentiate .
First, differentiate using the quotient rule:

Step 3: Substitute in the formula.
Now, use the derivative formula: Simplifying the term inside the square root:

Step 4: Conclusion.
Thus, the derivative is:

Step 1: Understand the function.
The function is , which is a linear function. For a function to be increasing, its derivative must be positive.

Step 2: Differentiate the function.
The derivative of with respect to is:

Step 3: Analyze the derivative.
Since is positive for all values of , the function is increasing for all .

Step 4: Conclusion.
Thus, the function is increasing on .

Step 1: Understand the function.
The function is , which is a power function where both the base and the exponent are . To differentiate this, we will use logarithmic differentiation.

Step 2: Apply logarithmic differentiation.
Let . Taking the natural logarithm of both sides: Using the logarithmic property , we get:

Step 3: Differentiate both sides.
Now differentiate both sides with respect to : On the left-hand side, by the chain rule: On the right-hand side, use the product rule:

Step 4: Solve for .
Now, multiply both sides by to get:

Step 5: Conclusion.
Thus, the derivative of with respect to is: