Derivatives of Power Functions

Solved Problems

Solution.

We rewrite the function as follows:

Using the constant multiple rule and the power rule, we have

Solution.

Differentiating as a power function with a fractional exponent, we have

Solution.

The derivative of this power function is given by

Solution.

This function can be represented as a polynomial:

Differentiating term by term, we obtain:

Solution.

First, we rewrite the function as follows:

Use the sum rule for the derivative:

Then we take out the constant factors and calculate the derivatives of the power functions:

Here we used the expression $$ $$ Simplifying, we have

Solution.

We turn to the expression in the power form:

The derivative of the difference of two functions is equal to the difference of the derivatives of these functions:

Calculating the derivatives of the power functions, we obtain:

Solution.

First we convert the terms of the function to power form:

Using the power rule, we obtain:

Solution.

We convert each term of the function into a power form:

Using the linear properties of the derivative and the power rule, we have

Solution.

Representing the terms in the form of power functions, we obtain the following expression for the derivative:

Solution.

Using the power rule, we get

Solution.

We convert the radical expressions to power form:

Then applying the power rule, we get

Solution.

Converting the function to a power form, we obtain:

Solution.

We convert the function to power form:

Apply the power rule:

Solution.

Similarly to the previous example, we have

Solution.

Differentiating this function as a power function, we obtain:

Solution.

We apply the perfect cube identity

Hence

Then using the basic differentiation rules and the power rule, we find the derivative: