by Diane Holcomb | Oct 10, 2024 | Calculus, Math

Derivatives of exponentials summary The derivative of $e^x$ The derivative of $e^x$ is $$frac{d}{dx} Big( e^x Big) = e^x.$$ We can prove this by going back to the definition. $$begin{eqnarray} frac{d}{dx} Big( e^x Big) &= lim_{hto 0} frac{e^{x+h}-e^x}{h}\ &...
by Diane Holcomb | Oct 8, 2024 | Calculus, Math

The antiderivative of 1/x Finding the integral of 1/x Because we can see that $frac{1}{x}$ can be written as $x^{-1}$ we’re always tempted to look at the power rule for integration to determine it’s integral, but we can see right away that this...
by Diane Holcomb | Sep 23, 2024 | Calculus, Math

Trig derivatives summary The derivative of sin x The derivative of $sin x$ is $$frac{d}{dx} left( sin x right) = cos x.$$ We can prove this by going back to the definition. $$frac{d}{dx} left( sin x right) = lim_{hto 0} frac{sin (x+h)-sin x}{h}.$$ The first thing we...
by Dogl | Sep 18, 2024 | Calculus, Math

When do we need the Chain Rule? We need the chain rule when our function $f(x)$ is actually a function nested inside of another function. For example $f(x) = sin (3x^2 +2)$ has an inner function $3x^2+2$ and an outer function $sin x$. In other words $f(x)$ is the...
by Dogl | Sep 13, 2024 | Calculus, Math

Integration by parts? What parts? Integration by parts comes from rewriting the product rule for derivatives and then integrating. So the “parts” are the fact that we’re going to work with two functions and treat them separately. Let’s remember...