The Definition of Derivative

Question

If f(x) = 3x+10

Find f'(6)

Answer 1

Here you take the derivative of f(x) using the power rule which states the derivative of any term of the form a*xⁿ will be (a*n)*x^n-1 and the sum rule for derivatives which states that the derivative of added terms is equivalent to adding the derivatives of each individual term.

In this function f(x) you see two terms 3x and 10 being added together, so we will have to take the derivative of each and add them to get the derivative of f(x)

Let's take the derivative of 3x first using the power rule:

In this case your coefficient is 3 (which is the a term in the expression described above) and your exponent on the variable x is 1 since x¹=x (this is the n term above).

So, using the expression described above we will multiply a*n = 3*1 = 3 to get the coefficient of the derivative and n-1 = 1-1 = 0 so your exponent on the x on the derivative is 0 and any number to the 0th power is 1: your derivative for the 3x term should therefore be 3*1 = 3

This will, in fact, be always the case when you have a constant expression multiplying a first degree variable (and variable without an exponent i.e. the exponent is 1). The derivative of this case would just be the constant coefficient (examples: derivative of 6*x is 6, derivative of 1982*x is 1982, derivative of -68*x is -68, etc.)

Let's now take the derivative of 10, a constant term with no variables:

In this case the a in the expression is 10 and the n in the xⁿ is 0 since 10*x⁰ = 10

So, using the power rule: a*n = 10*0 = 0. Since a*n is 0 and multiplies to the other term, we can conclude that the derivative is just equal to 0.

Whenever you take the derivative of a constant term (a term with no variables involved), the derivative will always equal 0.

Now, using the sum rule for derivatives we can add the derivative of 3x and 10 to get the derivative of f(x)

f'(x) = 3 + 0 = 3

f'(x) = 3

Now you plug a 6 in the function for all x's (in this case there are none so...)

f'(6) = 3

Answer 2

As usual, Mark M. is correct and on point.

It might be worth pointing out that the first derivative of a function is the slope of the line tangent to the graph of that function and as Mark correctly observes in the case of a linear function the slope is constant as is the derivative!

Answer 3

f(x) = 3x + 10 is a linear function (from Algebra 1).

Its slope (first derivative) is constant at 3.

f'(6) = 3