What is the derivative of $a^2$?

Question

My sum requires to find the derivative of $x^2+a^2$. So derivative of $x^2$ is $2x$ but what about $a^2$? Which formula to be applied a raise to $x$ or $x$ raise to $n$?

Answer 1

It depends what variable you are differentiating with respect to, and whether $a$ is a constant or a variable.

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With respect to $x$:

If $a$ is simply a constant, then $\frac{d(x^2+a^2)}{dx}=2x$, as the constant vanishes.

If $a$ is another variable, we use implicit differentiation: $\frac{d(x^2+a^2)}{dx}=2x+2a\frac{da}{dx}$

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With respect to $a$:

We treat $x$ as a variable so again with implicit differentiation: $\frac{d(x^2+a^2)}{da}=2x\frac{dx}{da}+2a$

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With respect to some other variable $t$:

Again we use implicit differentiation so: $\frac{d(x^2+a^2)}{dt}=2x\frac{dx}{dt}+2a\frac{da}{dt}$