Differential of a Function

Solved Problems

Solution.

We calculate the derivative using the product and chain rule:

The differential of the function is written as

Solution.

First we determine the derivative using the chain rule:

Apply the double angle identity

Hence

The differential of a function is written in the form

Then

Solution.

Substitute the values of $$ and $$ and calculate the differential $$

Solution.

Differentiate the given function:

At the point $$ the derivative is equal to

Hence, the differential of the function at this point is

Solution.

We calculate the differential of the function by the formula

The derivative is given by

At the initial point $$ the derivative is equal to

Calculate the differential of the independent variable:

Then the differential $$ is

The approximate value of the function at $$ is equal to

Solution.

First we compute the increment of the function:

At the same values of $$ and $$, the differential of the function is equal to

Solution.

Take the derivative of the function:

At the point $$ we have

Then

Solution.

Take the derivative:

At $$ the derivative is equal to

So the differential $$ at this point is

Solution.

Using the differentiation rules, we obtain:

Solution.

where $$ $$

Determine the derivative of the implicit function. Differentiating both sides with respect to $$ we obtain:

At the point $$, the derivative $$ is equal

The differential at this point is respectively written as

Solution.

We differentiate both sides of the equation with respect to $$ and find the derivative $$

Calculate the value of the derivative at the given point $$

The differential of the function at this point is written in the following form:

Solution.

Note that

We evaluate the differential $$ at the point $$ when $$ As

and

then

Similarly we can find that $$ for $$ Therefore, the force $$ is estimated to be in the range

Solution.

It is easy to see that

We estimate the approximate value of the force using the differential:

Take the derivative:

so when $$ the derivative is equal to

Assuming $$ we have

Similarly, we can find the differential for $$

Thus, the resistance force takes on values in the interval

Solution.

We calculate the corresponding values of the parameter $$ from the equation $$

Make sure that the value $$ satisfies the condition $$

When $$ the derivative has the following value:

Thus, the differential of the function at the point $$ is expressed by the formula

Solution.

First we determine the value of the parameter $$, which corresponds to the point $$ It follows from the second equation that

Check the value of $$ at $$

Find the derivative of the parametric function:

When $$ the derivative is respectively equal to

Consequently, the differential of the function at the point $$ has the form:

Solution.

We write the differential of the "outer" function:

Similarly, we find the differential of the "inner" function:

Substituting the expression for $$ in the previous formula, we obtain the differential $$ in invariant form: