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The chain rule

Differentiating a function of a function: outside derivative times inside derivative.

(f(g(x)))′ = f′(g(x)) · g′(x). Picture it: if y changes 3 times as fast as u, and u changes 2 times as fast as x, then y changes 6 times as fast as x. Think it: the derivative is a linear map, and the chain rule says the derivative of a composition is the composition of the derivatives — the same statement holds for Jacobians in any dimension.

ຕົວຢ່າງ​ທີ່​ໄດ້​ເຮັດ: derivative of sin(x^2)

Calculus · Derivatives

Differentiate sin(x^2)

\frac{d}{dx}\left[\sin{\left(x^{2} \right)}\right]

ຂັ້ນ​ຕອນ​ຕໍ່​ຂັ້ນ​ຕອນ

  1. \frac{d}{dx}\left[\sin{\left(x^{2} \right)}\right]

    Start from the derivative to compute.

  2. \cos{\left(x^{2} \right)} \frac{d}{d x} x^{2}

    Chain rule: (sin u)′ = cos u, times u′ where u = x^{2}.

  3. 2 x \cos{\left(x^{2} \right)}

    Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = 2.

ເປີດ​ເຜີຍ​ຄຳ​ຕອບ
f'(x) = 2 x \cos{\left(x^{2} \right)}

Symbols used here

\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
C,\ C_1,\ C_2
arbitrary constants
Constants of integration fixed by initial conditions.

How to: The chain rule

  1. Identify the outer function and the inner function u.
  2. Differentiate the outer function, leaving u alone.
  3. Multiply by the derivative of u.
  4. Simplify — and check by differentiating a small case numerically.

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

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