by Diane Holcomb | Nov 12, 2024 | Calculus, Math
Special limits summary Special trig limits There are two commonly used special Calculus limits involving trig functions. They are: $$lim_{xto 0} frac{sin x}{x} =1$$ and $$lim_{xto 0} frac{cos x – 1}{x} = 0.$$ These limit are necessary for determining the...
by Diane Holcomb | Oct 24, 2024 | Calculus, Math
Derivative Rules Linearity (or the sum rule) The derivative is linear, that is $$frac{d}{dx} Big(C f(x) + g(x) Big) = Cf'(x) + g'(x)$$ The Product and Quotient Rules $$frac{d}{dx} Big(f(x)g(x) Big) = f'(x)g(x) + f(x)g'(x)$$ $$frac{d}{dx} Big(frac{f(x)}{g(x)} Big) =...
by Diane Holcomb | Oct 14, 2024 | Calculus, Math
Derivatives of logarithms summary Derivative of $ln x$ The derivative of $ln x$ is$$frac{d}{dx} Big( ln x Big) = frac{1}{x}.$$We can prove this by using the fact that $ln x$ and $e^x$ are inverse functions and using the derivative of $e^x$, which is $e^x$. Therefore...
by Diane Holcomb | Oct 10, 2024 | Calculus, Math
Derivatives of exponentials summary The derivative of $e^x$ The derivative of $e^x$ is $$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...
