What is the derivative of f(x) = cos (x^2 - 4x)?

1 Answer
Aldrich
Sep 13, 2015

color(blue)(d/dx cos(x^2-4x)= -sin(x^2 - 4x) * (2x - 4))

Explanation:

To differentiate cos(x^2-4x), we have to apply the chain rule:

color(green)((f @ g)'(x) = f'(g(x)) * g'(x))

In other words:
1) Get the derivative of the outer function, and plug in the inner function...
2) Then multiply that by the derivative of the inner function.

In cos(x^2-4x), the outer function is cos x and the inner function is x^2 - 4x.

The derivative of cos x is -sin x, so we get:

d/dx cos(x^2-4x)

= -sin(x^2 - 4x) * d/dx (x^2 - 4x)

Next we solve d/dx (x^2 - 4x) using the power rule:

color(green)(d/dx x^n = nx^(n-1))

-sin(x^2 - 4x) * d/dx (x^2 - 4x)

= -sin(x^2 - 4x) * (2x - 4)

In summary:

d/dx cos(x^2-4x)

= -sin(x^2 - 4x) * d/dx (x^2 - 4x)

color(blue)(= -sin(x^2 - 4x) * (2x - 4))

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