How to Find the Derivative of a Constant Times a Function Using the Product Rule

derivative of a constant times a function


The derivative of a constant times a function can be found using the product rule of differentiation. Let’s say we have a constant, let’s call it “c”, and a function, let’s call it “f(x)”.

The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by:

(d/dx) [ u(x) * v(x) ] = u'(x) * v(x) + u(x) * v'(x)

In our case, “c” would be the constant and “f(x)” would be the function. Applying the product rule, we get:

(d/dx) [ c * f(x) ] = c * f'(x) + c’ * f(x)

Since c is a constant, its derivative c’ is zero. Therefore, the derivative of c times f(x) simplifies to:

(d/dx) [ c * f(x) ] = c * f'(x)

So, when you take the derivative of a constant multiplied by a function, you can simply bring out the constant and multiply it by the derivative of the function.

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