Question

How do you use the definition of a derivative to find the derivative of f(x) = – 3x?

Answer 1

Hint: We are given f(x) = – 3x and we are asked to find the derivative of f(x). To do the same, we will learn about the product rule. We will split our function f(x) = – 3x into two fractions and then apply the product rule where it is given as \[{{\left( uv \right)}^{'}}={{u}^{'}}v+{{v}^{'}}u\] and then we also need \[\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}\] to simplify our solution.

Complete step-by-step solution:
We are given a function f(x) = – 3x and we have to differentiate it. Now, we will first observe our function. We can see that f(x) = – 3x is given as the product of – 3 and x. So, we can see that our function is the product of 2 functions in which one is – 3 and the other is x. As we know to find the derivative of the product of 2 functions, we will need the product rule. The product rule is given as \[{{\left( uv \right)}^{'}}={{u}^{'}}v+{{v}^{'}}u.\] Now, in f(x) = – 3x, we consider u = – 3 and v = x. So, applying the product rule on f(x) = – 3x, we get,
\[\Rightarrow \dfrac{d\left( f\left( x \right) \right)}{dx}=\dfrac{d\left( -3x \right)}{dx}=x\dfrac{d\left( -3 \right)}{dx}+\left( -3 \right)\dfrac{d\left( x \right)}{dx}\]
Now, as we know the derivative of constant is zero, so \[\dfrac{d\left( -3 \right)}{dx}=0\] and we have \[\dfrac{dx}{dx}=1,\] as we compare \[\dfrac{d\left( x \right)}{dx}\] with \[\dfrac{d\left( {{x}^{n}} \right)}{dx}\] then we get n = 1 as \[\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}\] and \[\dfrac{dx}{dx}=1{{x}^{1-1}}=1.\]
Now using \[\dfrac{d\left( -3 \right)}{dx}=0\] and \[\dfrac{dx}{dx}=1,\] we get,
\[\Rightarrow \dfrac{d\left( f\left( x \right) \right)}{dx}=\dfrac{d\left( -3x \right)}{dx}=x\left( 0 \right)+\left( -3 \right)\left( 1 \right)\]
On simplifying, we get,
\[\Rightarrow \dfrac{d\left( -3x \right)}{dx}=-3\]
Here, the derivative of – 3x is – 3.

Note: As we can see that our function is the product of 2 functions of which one is constant, so there is another way to find the derivative. Our function is of the type f(x) = k.g(x), that is constant multiplied by other function and then in the derivative \[\dfrac{d\left( f\left( x \right) \right)}{dx}=\dfrac{d\left( kg\left( x \right) \right)}{dx},\] we can take out k and we get \[d\left( kg\left( x \right) \right)=kd\left( g\left( x \right) \right).\] So, in our case f(x) = – 3x.
\[\dfrac{d\left( -3x \right)}{dx}=-3\dfrac{dx}{dx}=-3\]
As \[\dfrac{dx}{dx}=1.\]
So, the derivative of f(x) = – 3x is – 3.