Derivative of cosh(x)
Answer
Derivative of cosh(x)
- Rule Used
- Hyperbolic Rule
- Steps
- Recognize a hyperbolic function. sinh and cosh swap under differentiation, with no minus sign, because both are built from e^x and with the same sign pattern.
- Final answer:
d/dx cosh(x) = sinh(x). The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
Instant step-by-step solution for cosh(x)
Derivative of cosh(x) = sinh(x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-cosh-x
Derivative
cosh(x) → sinh(x)engine: sinh(x)
Graph of the function and its derivative
The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.
Why the derivative has this form
cosh(x) = (e^x + e^(−x))/2. Differentiating gives (e^x − e^(−x))/2, which is sinh(x), with no minus in front. Cosine's derivative has a minus; cosh's does not, because the two exponential terms add in the definition and the minus from e^(−x) creates sinh rather than −sinh.
Apply the exponential rule to cosh(x). The form to remember, already simplified, is sinh(x).
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
At x = 0, substitute into the simplified form sinh(x). That number is the slope of cosh(x) there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = cosh(x).
- Apply the exponential rule to cosh(x). The form to remember, already simplified, is sinh(x).
- At x = 0, substitute into the simplified form sinh(x). That number is the slope of cosh(x) there; a secant over a small step in x lands next to it.
Common mistakes
Answering -sinh(x)
−sinh(x) imports the sign from cosine. Hyperbolic cosine is always at least 1 and its slope at zero is 0, but just to the right the slope is positive.
Copying a trigonometric minus onto cosh(x)
Hyperbolic sine and cosine come from exponentials. The minus signs inside those definitions do not leave a minus in front of the derivative the way cosine does.
Related rule and examples
- Exponential derivatives — The rule page lists this example and the others in the same family.
- Trigonometric derivatives — The rule page lists this example and the others in the same family.
- Derivatives of trigonometric functions — Same family of differentiation, different function.
- Derivative of sinh(x) — Same family of differentiation, different function.
- Derivative of tanh(x) — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of cosh(x)?
sinh(x). cosh(x) = (e^x + e^(−x))/2. Differentiating gives (e^x − e^(−x))/2, which is sinh(x), with no minus in front. Cosine's derivative has a minus; cosh's does not, because the two exponential terms add in the definition and the minus from e^(−x) creates sinh rather than −sinh.
Which wrong answer shows up for cosh(x)?
−sinh(x) imports the sign from cosine. Hyperbolic cosine is always at least 1 and its slope at zero is 0, but just to the right the slope is positive.
Where is the derivative of cosh(x) defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.