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# Prove that the derivative of cosh x is sinh x

Prove the following formula for the derivative of hyperbolic cosine,

Proof. We can compute from the definition of hyperbolic cosine in terms of exponentials,

# Prove that the derivative of sinh x is cosh x

Prove the following formula for the derivative of the hyperbolic sine,

Proof. We can compute the derivative from the definition of the hyperbolic sine in terms of exponentials,

# Calculate tanh (2x) given tanh x = 3/4

Find the value of given that .

We recall the following formulas (from Section 6.19, Exercises #7 and #8),

Then using the definition of hyperbolic tangent we compute

# Calculate cosh (x+y) given sinh x = 4/3 and sinh y = 3/4

Find the value of given that and .

We use the identity in both cases.

Then we recall (Section 6.19, Exercise #6) the formula for ,

Therefore, we can compute

# Calculate sinh x and cosh x given tanh x = 5/13

Find the values of and given that .

Since (see here, Section 6.19 Exercise #14) we have

Then, to compute we have,

# Calculate sinh x given cosh x = 5/4 and x > 0

Find the value of given that and .

We know . So, if we have

Thus, if ,

# Calculate cosh x when sinh x = 4/3

Find the value of when given .

We know . So, if then we have

Hence,

# Prove that coth2 x – csch2 x = 1

Prove the following identity,

Proof. From the definitions of hyperbolic cotangent and hyperbolic cosecant we have,

# Prove that tanh2 x + sech2x = 1

Prove the following identity,

From the definitions of hyperbolic tangent and hyperbolic secant we have,

# Prove that 2 cosh2 (x/2) = cosh x + 1

Prove the following identity,

Proof. Computing directly from the definition of the hyperbolic cosine,