## Derivative of a Sum

When calculating the derivative of a sum, we simply take the sum of the derivatives. This is illustrated in the following formula.

The first function that your boss wants you to work with is the sum of two functions. Therefore, to find the derivative of this function, we just take the sum of the derivatives. To do this, we need to recognize that the derivative of *x* 2 is 2*x*, and the derivative of 4*x* is 4. Now we just plug into our formula and we've got our derivative.

We see that the derivative of the first function is *f* ' (*x*) = 2*x* + 4.

## Derivative of a Difference

The second function, *g*, is a difference of functions. When we want to find the derivative of a difference, we simply find the difference of the derivatives. This is similar to the sum rule, and is shown in the following formula.

To use our formula for our example, we just need to know that the derivative of 3*x*3 is 9*x* 2, and the derivative of 5*x* 2 is 10*x*. Once again, we just plug into our formula.

Using our formula, we've found that the derivative of the second function is *g*' (*x*) = 9*x* 2 - 10*x*.

## Derivative of a Product

Now things get a little trickier. The third function is a product of the functions *x* and ln(*x*). When we want to find the derivative of a product, we use the product rule for derivatives.

Thus, to find the derivative of our function *h* in our example, we need to know that the derivative of *x* is 1, and the derivative of ln(*x*) is 1/*x*, and then we just need to use our formula.

We see that the derivative of the third function is *h*' (*x*) = ln(*x*) + 1.

## Derivative of a Quotient

The last function is a quotient. Once again, taking the derivative of a quotient is a little trickier than just taking the quotient of the derivatives. We have a quotient rule for derivatives as well, and it is as follows.

To use the quotient rule to find the derivative of *j*, we need to know the derivative of *x* and ln(*x*). We actually already know these from the last problem. The derivative of *x* is 1, and the derivative of ln(*x*) is 1/*x*. All we have to do now is plug into our formula and simplify.

We've found the derivative of the last function to be *j* ' (*x*) = (1 - ln(*x*)) / *x* 2.

We've now found all the derivatives of the functions that your boss gave you, so now he can find the slope of each of those functions at any given value of *x*. Great job! You should ask for a raise!

## Lesson Summary

When it comes to finding the derivative of a sum, difference, product, or quotient, we have different rules for each case. These rules are as follows.

When we want to use one of these formulas, we simply calculate the different parts of the formula and then plug in and simplify. We can see that these formulas make finding these derivatives fairly easy, so it is a good idea to try to put these to memory for future use.