# Thread: Differentiating both sides of an equation

1. ## Differentiating both sides of an equation

Hi guys,

When is it justified to differentiate both sides of an equation? I'm just reading a textbook, which out of the blue takes the derivative of both sides, without any real justification. For instance, I can't take the derivative of $5x = x^2 + 5x -4$, and still preserve that equality.

HTale.

2. Originally Posted by HTale
Hi guys,

When is it justified to differentiate both sides of an equation? I'm just reading a textbook, which out of the blue takes the derivative of both sides, without any real justification. For instance, I can't take the derivative of $5x = x^2 + 5x -4$, and still preserve that equality.

HTale.
diffferentiation is used to find the instantaneous rate of change.

if you differentiate the equation $5x = x^2 + 5x - 4$ , then you are determining a new equation that tell you where the slope of the two graphs are equal.

$\frac{d}{dx}(5x = x^2 + 5x - 4)$

$
5 = 2x + 5
$

$x = 0$ is the location where the slope of $y = 5x$ and the slope of $y = x^2 + 5x - 4$ are equal.

3. If two functions f(x) and g(x) are identically equal then you can differentiate both sides of the equation f(x) = g(x), and the resulting equation f'(x) = g'(x) will also be identically true. For example, $\sin 2x = 2\sin x\cos x$ (for all x). Differentiate both sides and you get $2\cos2x = 2\cos^2x-2\sin^2x$, also true for all x.

But the equation $5x = x^2+5x-4$ is not an identity. It is only true for two values of x. When you differentiate it you get a completely different equation, which has a different interpretation, as skeeter has pointed out.