d/dx [kf(x)]=
kf'(x)
The derivative of a constant k with respect to x is zero since the constant does not depend on x. Therefore, we can apply the chain rule to find the derivative of kf(x) with respect to x.
d/dx [kf(x)] = k[d/dx f(x)]
The derivative of f(x) with respect to x is multiplied by the constant k. Thus, the derivative of kf(x) with respect to x is:
d/dx [kf(x)] = k[d/dx f(x)]
Alternatively, we could use the product rule to find the derivative of kf(x). If we let u(x) = k and v(x) = f(x), then du/dx = 0 and dv/dx = d/dx f(x). Using the product rule,
d/dx [u(x)v(x)] = u(x) * dv/dx + v(x) * du/dx
Substituting u(x) and v(x) gives:
d/dx [kf(x)] = k * d/dx f(x)
Both of these approaches give the same result, which is that the derivative of kf(x) with respect to x is k times the derivative of f(x) with respect to x.
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