Chain Rule Patterns Reference: Trig, Exp, Log, and Power

June 1, 2026

How to Use This Reference

Every pattern below applies the same chain rule formula: differentiate the outer function (keeping the inner function as its argument), then multiply by the derivative of the inner function.

In each case, g(x) represents any differentiable inner function. g’(x) is its derivative.

Use the Chain Rule Calculator to apply these patterns step by step for any specific function pair. For the full formula derivation and notation explanation, see the Chain Rule Formula Reference.

Power Functions

Pattern: d/dx[g(x)ⁿ] = n · g(x)^(n−1) · g’(x)

Expression	Inner g(x)	g’(x)	Result
(x² + 1)^5	x² + 1	2x	10x(x² + 1)^4
(sin x)^3	sin x	cos x	3cos(x)·sin²(x)
(3x + 2)^−2	3x + 2	3	−6(3x + 2)^−3
√(x² + 4) = (x² + 4)^(1/2)	x² + 4	2x	x/√(x² + 4)
(ln x)^4	ln x	1/x	4(ln x)³ / x

Note on square roots: Always rewrite √g(x) as g(x)^(1/2) before applying the power pattern. Then d/dx[g(x)^(1/2)] = (1/2)g(x)^(−1/2) · g’(x) = g’(x) / (2√g(x)).

Trigonometric Functions

Sine

Pattern: d/dx[sin(g(x))] = cos(g(x)) · g’(x)

Expression	g(x)	g’(x)	Result
sin(x²)	x²	2x	2x·cos(x²)
sin(3x)	3x	3	3cos(3x)
sin(e^x)	e^x	e^x	e^x·cos(e^x)
sin(√x)	√x	1/(2√x)	cos(√x) / (2√x)

Cosine

Pattern: d/dx[cos(g(x))] = −sin(g(x)) · g’(x)

Expression	g(x)	g’(x)	Result
cos(x³)	x³	3x²	−3x²·sin(x³)
cos(2x + 1)	2x + 1	2	−2sin(2x + 1)
cos(ln x)	ln x	1/x	−sin(ln x) / x

Tangent

Pattern: d/dx[tan(g(x))] = sec²(g(x)) · g’(x)

Expression	g(x)	g’(x)	Result
tan(x²)	x²	2x	2x·sec²(x²)
tan(5x)	5x	5	5sec²(5x)

Secant, Cosecant, Cotangent

Function	Pattern
sec(g(x))	sec(g(x)) · tan(g(x)) · g’(x)
csc(g(x))	−csc(g(x)) · cot(g(x)) · g’(x)
cot(g(x))	−csc²(g(x)) · g’(x)

Exponential Functions

Natural Exponential e^x

Pattern: d/dx[e^(g(x))] = e^(g(x)) · g’(x)

Expression	g(x)	g’(x)	Result
e^(3x)	3x	3	3e^(3x)
e^(x²)	x²	2x	2x·e^(x²)
e^(sin x)	sin x	cos x	cos(x)·e^(sin x)
e^(−x)	−x	−1	−e^(−x)
e^(√x)	√x	1/(2√x)	e^(√x) / (2√x)

General Base aˣ

Pattern: d/dx[a^(g(x))] = a^(g(x)) · ln(a) · g’(x)

Expression	Result
2^(3x)	2^(3x) · ln(2) · 3 = 3·ln(2)·2^(3x)
10^(x²)	10^(x²) · ln(10) · 2x

Logarithmic Functions

Natural Logarithm

Pattern: d/dx[ln(g(x))] = g’(x) / g(x)

Expression	g(x)	g’(x)	Result
ln(x² + 1)	x² + 1	2x	2x / (x² + 1)
ln(sin x)	sin x	cos x	cos(x) / sin(x) = cot(x)
ln(3x)	3x	3	3/(3x) = 1/x
ln(e^x + 1)	e^x + 1	e^x	e^x / (e^x + 1)

Useful shortcut: d/dx[ln(ax)] = 1/x for any constant a (the constant cancels).

General Logarithm Base a

Pattern: d/dx[log_a(g(x))] = g’(x) / (g(x) · ln(a))

Expression	Result
log₂(x²)	2x / (x² · ln 2) = 2 / (x · ln 2)
log₁₀(3x + 1)	3 / ((3x + 1) · ln 10)

Inverse Trigonometric Functions

Arcsine

Pattern: d/dx[arcsin(g(x))] = g’(x) / √(1 − g(x)²)

Expression	Result
arcsin(2x)	2 / √(1 − 4x²)
arcsin(x²)	2x / √(1 − x⁴)

Arccosine

Pattern: d/dx[arccos(g(x))] = −g’(x) / √(1 − g(x)²)

Note: arccos derivative is the negative of arcsin.

Arctangent

Pattern: d/dx[arctan(g(x))] = g’(x) / (1 + g(x)²)

Expression	Result
arctan(3x)	3 / (1 + 9x²)
arctan(x²)	2x / (1 + x⁴)
arctan(e^x)	e^x / (1 + e^(2x))

Nested Compositions (Multiple Chain Rule)

When functions are nested three or more levels deep, apply the chain rule repeatedly from outside in.

Triple composition: d/dx[f(g(h(x)))] = f’(g(h(x))) · g’(h(x)) · h’(x)

Expression	Layers	Result
sin(e^(x²))	sin → e → x²	cos(e^(x²)) · e^(x²) · 2x
e^(sin(3x))	e → sin → 3x	e^(sin(3x)) · cos(3x) · 3
ln((x² + 1)^3)	ln → power → x²+1	3·2x/(x²+1) = 6x/(x²+1)
√(sin(x²))	sqrt → sin → x²	cos(x²)·2x / (2√(sin(x²)))

For nested compositions, the Chain Rule Formula Reference covers the general multi-layer approach. To practice identifying which pattern applies, work through the Chain Rule Practice Problems.

Pattern Summary Table

Outer function	d/dx[outer(g(x))]
g(x)^n	n·g(x)^(n−1)·g’(x)
√g(x)	g’(x) / (2√g(x))
sin(g(x))	cos(g(x))·g’(x)
cos(g(x))	−sin(g(x))·g’(x)
tan(g(x))	sec²(g(x))·g’(x)
e^(g(x))	e^(g(x))·g’(x)
ln(g(x))	g’(x)/g(x)
arcsin(g(x))	g’(x)/√(1−g(x)²)
arctan(g(x))	g’(x)/(1+g(x)²)