Differentiating Hyperbolic Functions

Differentiating hyperbolic functions is straightforward from the exponential definitions. The results mirror the trigonometric derivatives but with important sign differences.

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1 Standard Derivatives

d/dx (sinh x) = cosh x

d/dx (cosh x) = sinh x

d/dx (tanh x) = sech^2 x

Compare: d/dx(sin x) = cos x, d/dx(cos x) = **-**sin x, d/dx(tan x) = sec^2 x.

The key difference: d/dx(cosh x) = sinh x (positive), not negative.

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2 Proofs

d/dx (sinh x)

d/dx [(e^x - e^(-x))/2] = (e^x + e^(-x))/2 = cosh x. QED.

d/dx (cosh x)

d/dx [(e^x + e^(-x))/2] = (e^x - e^(-x))/2 = sinh x. QED.

d/dx (tanh x)

Quotient rule on sinh x / cosh x: (cosh^2 x - sinh^2 x)/cosh^2 x = 1/cosh^2 x = sech^2 x. QED.

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3 Reciprocal Function Derivatives

d/dx (sech x) = -sech x tanh x

d/dx (cosech x) = -cosech x coth x

d/dx (coth x) = -cosech^2 x

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4 Chain Rule Applications

d/dx (sinh(f(x))) = cosh(f(x)) * f'(x)

d/dx (cosh(f(x))) = sinh(f(x)) * f'(x)

d/dx (tanh(f(x))) = sech^2(f(x)) * f'(x)

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5 Worked Example 1

Differentiate y = sinh(3x^2).