Derivatives of common functions
The product, quotient and chain rules are the rules that combine functions; now you need to know how to differentiate each type of basic function: exponentials, logarithms, sines, cosines and their inverses. With a fixed table of derivatives and the chain rule, you can differentiate practically any function you will meet in Bachillerato.
Exponential and logarithmic functions
Find the derivative of the following composite exponential function: $q(x) = e^{4x^3 - 7x}$
Find the derivative of the following composite exponential function:
$q(x) = e^{4x^3 - 7x}$
Determine the derivative of the following composite logarithmic function: $s(x) = \ln(2x^4 + x^2 - 3)$
Determine the derivative of the following composite logarithmic function:
$s(x) = \ln(2x^4 + x^2 - 3)$
Trigonometric functions
Calculate the derivative of the following trigonometric function: $f(x) = 5\sin x - 3\cos x + \tan x$
Calculate the derivative of the following trigonometric function:
$f(x) = 5\sin x - 3\cos x + \tan x$
Repeatedly apply the chain rule to find the derivative of the following trigonometric function: $f(x) = \sin^3(2x^2 + \pi)$
Repeatedly apply the chain rule to find the derivative of the following trigonometric function:
$f(x) = \sin^3(2x^2 + \pi)$
Powers and roots of a function (chain rule)
Apply the chain rule to differentiate the following power of a function: $p(x) = (3x^2 - 5x + 1)^4$
Apply the chain rule to differentiate the following power of a function:
$p(x) = (3x^2 - 5x + 1)^4$
Calculate the derivative of the function involving a radical: $r(x) = \sqrt{5x^2 - 3x + 2}$
Calculate the derivative of the function involving a radical:
$r(x) = \sqrt{5x^2 - 3x + 2}$
Inverse trigonometric functions
Calculate the derivative of the following function that combines inverse trigonometric functions with exponentials and polynomials: $h(x) = \arctan(e^x) + \arcsin(2x)$
Calculate the derivative of the following function that combines inverse trigonometric functions with exponentials and polynomials:
$h(x) = \arctan(e^x) + \arcsin(2x)$
Product and quotient with common functions
The functions in the table almost never appear on their own: they are combined using products and quotients. The strategy does not change: identify the outer operation and apply the corresponding rule, using the table for each factor.
Apply the product rule to find the derivative of the function: $g(x) = x^3 \ln x$
Apply the product rule to find the derivative of the function:
$g(x) = x^3 \ln x$
Find the derivative of the following quotient of functions: $h(x) = \frac{e^x}{x^2+1}$
Find the derivative of the following quotient of functions:
$h(x) = \frac{e^x}{x^2+1}$
The position of a mass attached to a damped spring is given by the function $x(t) = 5 e^{-0,4t} \cos(3t)$, where $x$ is measured in centimetres and $t$ in seconds.…
The position of a mass attached to a damped spring is given by the function $x(t) = 5 e^{-0,4t} \cos(3t)$, where $x$ is measured in centimetres and $t$ in seconds. Determine the velocity function $v(t)$ of the oscillator by differentiating the position function.