How to Find the Derivative of the Product of Two Functions using the Product Rule

d/dx (k×f)=

To find the derivative of the product of two functions, k and f, with respect to x, we can use the product rule

To find the derivative of the product of two functions, k and f, with respect to x, we can use the product rule. The product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

In this case, we are finding the derivative of k times f with respect to x, so we can write it as:

d/dx (k×f) = k × d/dx(f) + f × d/dx(k)

Now, let’s consider the derivatives of the individual functions k and f:

d/dx(k) represents the derivative of k with respect to x. If k is a constant, then its derivative will be zero since the rate of change of a constant function is always zero.

d/dx(f) represents the derivative of f with respect to x.

Now, let’s substitute these derivatives back into the initial equation:

d/dx (k×f) = k × d/dx(f) + f × d/dx(k)

Since k is a constant, its derivative is zero:

d/dx (k×f) = 0 × f + f × d/dx(k)

Simplifying further:

d/dx (k×f) = 0 + f × d/dx(k)

Therefore, the derivative of k times f with respect to x is equal to f times the derivative of k:

d/dx (k×f) = f × d/dx(k)

It’s important to note that this result assumes that k and f are functions of x. If they are functions of other variables, the derivative will be different.

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