2.3.6 Practice set: differentiation

Power, constant and sum rules

Power, constant and sum rules (lesson 2.3.1)

0 of 6 correct

1.
Differentiate $x^8$
2.
Differentiate $3x^4 - x + 6$
3.
Differentiate $x^{3/2}$
4.
Differentiate $\dfrac{2x^3 + 1}{x}$
5.
Find the slope of $x^3 - 3x$ at $x = -1$
6.
Find the slope of $\dfrac{1}{x^2}$ at $x = 1$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Product and quotient rules

Product and quotient rules (lesson 2.3.2)

0 of 6 correct

1.
Differentiate $(x + 1)(x^2 - 2)$ and expand
2.
Differentiate $\dfrac{x}{x + 2}$
3.
Find the slope of $\sqrt{x}\,(x + 3)$ at $x = 1$
4.
At $x = 1$ : $f = 2$ , $g = -1$ , $f' = 3$ and $g' = 4$ . Find the slope of $\frac{f}{g}$ at 1
5.
Differentiate $\dfrac{x - 3}{x + 3}$
6.
Find the slope of $x^2(x + 1)$ at $x = 2$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

The chain rule

The chain rule (lesson 2.3.3)

0 of 6 correct

1.
Differentiate $(2x + 3)^4$
2.
Differentiate $\sqrt{4x + 1}$
3.
Differentiate $(1 - x^2)^3$
4.
Find the slope of $\big((2x - 1)^2 + 3\big)^2$ at $x = 1$
5.
Differentiate $x^2(x - 1)^3$
6.
Find the slope of $(x^2 + 1)^4$ at $x = 1$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Exponentials and logarithms

Exponentials and logarithms (lesson 2.3.4)

0 of 6 correct

1.
Differentiate $e^{-0.05x}$
2.
Find the slope of $e^{2x}$ at $x = 0$
3.
Differentiate $x\ln x$
4.
Differentiate $\ln(1 + e^x)$
5.
Differentiate $10^x$
6.
Differentiate $\ln(3 + x)$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Sine and cosine

Sine and cosine (lesson 2.3.5)

0 of 6 correct

1.
Find the slope of $\sin x$ at $x = \frac{\pi}{6}$ (type sqrt for a square root)
2.
Find the slope of $\cos(5x)$ at $x = \frac{\pi}{10}$
3.
Find the slope of $\tan(2x)$ at $x = \frac{\pi}{8}$
4.
Find the second derivative of $\cos(3x)$ at $x = 0$
5.
With $x$ in degrees, find the slope of $\cos x$ at $x = 90$ , per degree (type pi for $\pi$ )
6.
Find the slope of $\cos^2 x$ at $x = \frac{\pi}{4}$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

A derivative checker

Write is_derivative(f, claimed). Both arguments are strings in Python syntax, such as "x**3" and "3*x**2". Return True if claimed is the derivative of f with respect to x, and False otherwise.

sp.sympify(text) turns a string into a SymPy expression; sp.symbols("x") gives the letter $x$ .
Two expressions are equal when sp.simplify(a - b) == 0.

The tests include right answers written in unusual forms. They also include the mistakes from these lessons: a missing chain factor, $f'g$ for a product, a quotient rule with its top swapped, and the wrong sign for the slope of $\cos x$ .

See the tests

def test_power_rule():
    """3x² is the derivative of x³"""
    assert is_derivative("x**3", "3*x**2") is True, 'expected True for ("x**3", "3*x**2")'


def test_unusual_form():
    """A right answer in another form still counts: (3x + 1)⁵ → 15(3x + 1)⁴ written expanded"""
    expanded = "1215*x**4 + 1620*x**3 + 810*x**2 + 180*x + 15"
    assert is_derivative("(3*x + 1)**5", expanded) is True, "expected True for the expanded form"


def test_forgot_chain_factor():
    """Forgetting the chain rule's inner factor is wrong: (3x + 1)⁵ → 5(3x + 1)⁴"""
    assert is_derivative("(3*x + 1)**5", "5*(3*x + 1)**4") is False, "expected False: the factor 3 is missing"


def test_product_mistake():
    """(fg)' is not f'g: x²·eˣ → 2x·eˣ is wrong"""
    assert is_derivative("x**2*exp(x)", "2*x*exp(x)") is False, "expected False for f'g alone"


def test_quotient_order():
    """Swapping the top of the quotient rule is wrong: x/(x + 1) → -1/(x + 1)²"""
    assert is_derivative("x/(x + 1)", "-1/(x + 1)**2") is False, "expected False: the right answer is 1/(x + 1)²"


def test_chain_with_exp():
    """A chain rule answer with an exponential: e^(-x²/2) → -x·e^(-x²/2)"""
    assert is_derivative("exp(-x**2/2)", "-x*exp(-x**2/2)") is True, 'expected True for ("exp(-x**2/2)", "-x*exp(-x**2/2)")'


def test_logs():
    """ln(5x) has derivative 1/x"""
    assert is_derivative("log(5*x)", "1/x") is True, 'expected True for ("log(5*x)", "1/x")'


def test_cosine_sign():
    """The slope of cos x is −sin x, so sin x is wrong"""
    assert is_derivative("cos(x)", "sin(x)") is False, 'expected False for ("cos(x)", "sin(x)"): the slope of cos x is -sin x'

Power, constant and sum rules

Product and quotient rules

The chain rule

Exponentials and logarithms

Sine and cosine

Coding exercise

Free further reading