Derivative + Integral Formulas

d/dx [k]

0

d/dx [f(x) ± g(x)]

f′(x) ± g′(x)

d/dx [k·f(x)]

k·f′(x)

d/dx [f(x)·g(x)]

g(x)·f′(x) + g’(x)·f(x)

d/dx [f(x)/g(x)]

[g(x)·f′(x) − f(x)·g′(x)] / [g(x)]² (domain: g(x) ≠ 0)

d/dx [f(g(x))]

f′(g(x)) · g′(x) (chain rule; domain: g(x) in domain of f)

d/dx [xⁿ]

n · xⁿ⁻¹

d/dx [uⁿ] (chain form)

n · uⁿ⁻¹ · u′ (domain: depends on ‘n’; for non-integer ‘n’ ensure ‘u’ stays in domain)

d/dx [sin(u)]

cos(u) · u′ (domain: all real u)

d/dx [cos(u)]

−sin(u) · u′ (domain: all real u)

d/dx [tan(u)]

sec²(u) · u′ (domain: u ≠ π/2 + kπ)

d/dx [cot(u)]

−csc²(u) · u′ (domain: u ≠ kπ)

d/dx [sec(u)]

sec(u)·tan(u) · u′ (domain: u ≠ π/2 + kπ)

d/dx [csc(u)]

−csc(u)·cot(u) · u′ (domain: u ≠ kπ)

d/dx [e^^{u}]

e^^{u} · u′ (domain: all real u)

d/dx [a^^{u}]

a^^{u}·ln(a) · u′ (domain: a>0)

d/dx [ln|u|]

u′ / u (domain: u ≠ 0)

d/dx [sin⁻¹(u)]

u′ / √(1 − u²) (domain: |u| < 1)

d/dx [cos⁻¹(u)]

− u′ / √(1 − u²) (domain: |u| < 1)

d/dx [tan⁻¹(u)]

u′ / (1 + u²) (domain: all real u)

d/dx [cot⁻¹(u)]

− u′ / (1 + u²) (domain: all real u)

d/dx [sec⁻¹(u)]

u′ / (|u|·√(u² − 1)) (domain: |u| > 1)

d/dx [csc⁻¹(u)]

− u′ / (|u|·√(u² − 1)) (domain: |u| > 1)

d/dx [log__a(u)]

u’/u·​​​ln(a) (domain: a > 0, a ≠ 1, u>0)

f’(c) (Use for specific ‘c’ value)

\lim_{x \to c} \frac{f(x) - f(c)}{x - c}

How do you check if a function is differentiable at x=c using the limit definition of the derivative?

1. Compute the left-hand derivative:
   f'_-(c) = \lim_{x \to c^-} \frac{f(x) - f(c)}{x - c}

2. Compute the right-hand derivative:
   f'_+(c) = \lim_{x \to c^+} \frac{f(x) - f(c)}{x - c}

3. Compare:

   • If f'_-(c) = f'_+(c), f is differentiable at c.

   • If they are not equal, f is not differentiable at c .

If f is differentiable at x = c, then…

f is continuous at x = c
However, the opposite is not always true.

Limit Definition of a Derivative

\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}

Where is a function not differentiable?

A function is not differentiable at points where it is not continuous, has a corner or cusp, a vertical tangent, or where the left-hand and right-hand derivatives are not equal.

Left-Sum

Σ(i=1 to n) f(a + (i-1)Δx) · Δx where Δx = (b-a)/n

Right-Sum

Σ(i=1 to n) f(a + iΔx) · Δx where Δx = (b-a)/n

Midpoint-Sum

Σ(i=1 to n) f(a + (i - 1/2)Δx) · Δx where Δx = (b-a)/n

Trapezoidal-Sum

Σ(i=1 to n) [(f(x_{i-1}) + f(x_i))/2] · Δx where Δx = (b-a)/n

∫(a to b) f(x)dx

F(b) - F(a)

Mean Value Theorem for Integrals

If f is continuous on [a,b], then there exists a number c in [a,b] such that… ∫(a to b) f(x)dx = f(c) · (b-a)

[Where f(c) is known as the average value of the function]

d/dx[∫(a to x) f(t)dt]

d/dx[∫(a to x) f(t)dt] = f(x)

d/dx[∫(a to u(x)) f(t)dt]

d/dx[∫(a to u(x)) f(t)dt] = f(u(x)) · u'(x)

Average Value

f(c) = (1/(b-a)) · ∫(a to b) f(x)dx

∫[1/((x-a)(x-b))]dx

∫[1/((x-a)(x-b)(x-c))]dx

∫[1/((x-a)(x-b))]dx = ∫[A/(x-a) + B/(x-b)]dx

∫[1/((x-a)(x-b)(x-c))]dx = ∫[A/(x-a) + B/(x-b) + C/(x-c)]dx

∫(1/x)dx

∫(1/u)du

∫(u'/u)dx

∫(1/x)dx = ln|x| + C

∫(1/u)du = ln|u| + C

∫(u'/u)dx = ln|u| + C

∫sin(u)du

-cos(u) + C

∫cos(u)du

sin(u) + C

∫tan(u)du

-ln|cos(u)| + C or ln|sec(u)| + C

∫cot(u)du

ln|sin(u)| + C

∫sec(u)du

ln|sec(u) + tan(u)| + C

∫csc(u)du

-ln|csc(u) + cot(u)| + C or ln|csc(u) - cot(u)| + C

∫(1/√(a² - u²))du

arcsin(u/a) + C

∫(1/√(a² - u²))du

-arccos(u/a) + C

∫(1/(a² + u²))du

(1/a)arctan(u/a) + C

∫(1/(a² + u²))du

-(1/a)arccot(u/a) + C

∫(1/(u√(u² - a²)))du

(1/a)arcsec(|u|/a) + C

∫(1/(u√(u² - a²)))du

-(1/a)arccsc(|u|/a) + C

Integration by Parts

∫udv=uv−∫vdu

Pythagorean Identities

sin²(x) + cos²(x) = 1

sec²(x) = 1 + tan²(x)

1 + cot²(x) = csc²(x)

Power Reducing Formulas

sin²(x) = (1 - cos(2x))/2

cos²(x) = (1 + cos(2x))/2

Strategy with Integrals Involving Sine and Cosine.

\int_{}^{}\!\sin^{m}\left(x\right)\cos^{n}\left(x\right)\,dx

Strategy:

• Break up the Trig function that has the odd power and use the Pythagorean Identity sin²(x) + cos²(x) = 1.

• If there are no odd powers of sine and cosine, convert sin²(x) or cos²(x) using the Power Reducing Formulas sin²(x) = (1 - cos(2x))/2 or cos²(x) = (1 + cos(2x))/2.

Strategy with Integrals Involving Secant and Tangent.

\int_{}^{}\!\,\sec^{m}\left(x\right)\tan^{n}\left(x\right)dx

Strategy:

• If the power of secant is even, save a sec²(x) and convert the rest to tangents.

• If the power of tangent is odd, save a secant-tangent factor and convert the rest to secants.

• If there is just tangent to an even power, convert tan²(x) to sec²(x) - 1 and expand.

• If there is just secant to an odd power, use integration by parts.

• If none of the above apply, convert to sines and cosines.

Integrals with Infinite Limits of Integration

If f(x) is continuous on [a,\infty), then\int_{a}^{\infty}\!f\left(x\right)\,dx = limit (b →\infty) \int_{a}^b\!f\left(x\right)\,dx

If f(x) is continuous on [-\infty, b], then\int_{-\infty}^{b\!}f\left(x\right)\,dx = limit (a → -\infty)\int_{a}^b\!f\left(x\right)\,dx

If f(x) is continuous on (-\infty ,\infty), then\int_{-\infty}^{\infty}f\left(x\right)\,dx = \int_{-\infty}^{c}\!f\left(x\right)\,dx + \int_{c}^{\infty}\!f\left(x\right)\,dx where c is any real number.

Integrals with Infinite Discontinuities

If f(x) is continuous on (a, b], then\int_{a}^{b}\!f\left(x\right)\,dx = limit (c → a⁺) \int_{c}^{b}\!f\left(x\right)\,dx

If f(x) is continuous on [a, b), then\int_{a}^{b\!}f\left(x\right)\,dx = limit (c → b^-)\int_{a}^{c}\!f\left(x\right)\,dx

If f(x) is continuous on [a, b] but there is a vertical asymptote at c, then\int_{a}^{b}f\left(x\right)\,dx = \int_{a}^{c}\!f\left(x\right)\,dx + \int_{c}^{b}\!f\left(x\right)\,dx

Area Between Curves

If f and g are continuous on [a,b] and g(x) \le f(x) for all x in [a,b], then the area of the region bounded by the graphs of f and g on the interval [a,b] is given by…

A =\int_{a}^{b}\left(f\left(x\right)-g\left(x\right)\right)\!\,dx