Derivative exercises
You already know all the pieces: the power rule, the product, quotient and chain rules, and the table of exponential, logarithmic, trigonometric and inverse functions. What is still missing is the fluency to combine them without hesitation, faced with any function, deciding at first glance which rule takes charge and in what order the others are applied.
Deciding which rule to apply first
Find the derivative of the function $f(x) = (2x^2 + 3) e^x$.
Find the derivative of the function $f(x) = (2x^2 + 3) e^x$.
Determine the derivative of the rational function $f(x) = \frac{3x - 1}{x^2 + 4}$.
Determine the derivative of the rational function $f(x) = \frac{3x - 1}{x^2 + 4}$.
Chains of chains
Some functions need the chain rule several times in a row: a power of a sine of an exponential, for example. You differentiate from the outside in, layer by layer, without skipping any.
Calculate the derivative of the function $f(x) = (5x^3 - 2x)^4$ using the chain rule.
Calculate the derivative of the function $f(x) = (5x^3 - 2x)^4$ using the chain rule.
Find the derivative of the trigonometric function $f(x) = \cos(x^3 - 4x)$.
Find the derivative of the trigonometric function $f(x) = \cos(x^3 - 4x)$.
Mixing exponentials, logarithms and powers with a different base
Calculate the derivative of the function $f(x) = \ln(7x^2 + 5) + 3^{2x}$.
Calculate the derivative of the function $f(x) = \ln(7x^2 + 5) + 3^{2x}$.
Find the derivative of the irrational function $f(x) = \sqrt{x^4 - 3x^2 + 2}$.
Find the derivative of the irrational function $f(x) = \sqrt{x^4 - 3x^2 + 2}$.
Simplifying the result
Calculate the derivative of the function $f(x) = \frac{\sqrt{x^2-1}}{x+2}$ and simplify the result as much as possible, expressing it with a single denominator.
Calculate the derivative of the function $f(x) = \frac{\sqrt{x^2-1}}{x+2}$ and simplify the result as much as possible, expressing it with a single denominator.
Calculate the derivative of the composite function $f(x) = \sin^3(e^{2x} + 1)$, taking care with the successive application of the chain rule.
Calculate the derivative of the composite function $f(x) = \sin^3(e^{2x} + 1)$, taking care with the successive application of the chain rule.