Derivative of sinh(x)
Answer
Derivative of sinh(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 sinh(x) = cosh(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 sinh(x)
Derivative of sinh(x) = cosh(x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-sinh-x
Derivative
sinh(x) → cosh(x)engine: cosh(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
sinh(x) = (e^x − e^(−x))/2. Differentiating term by term gives (e^x − (−e^(−x)))/2 = (e^x + e^(−x))/2, which is cosh(x). There is no minus in the final answer, unlike cosine. The minus inside the definition is canceled by the derivative of e^(−x).
Apply the exponential rule to sinh(x). The form to remember, already simplified, is cosh(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 cosh(x). That number is the slope of sinh(x) there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = sinh(x).
- Apply the exponential rule to sinh(x). The form to remember, already simplified, is cosh(x).
- At x = 0, substitute into the simplified form cosh(x). That number is the slope of sinh(x) there; a secant over a small step in x lands next to it.
Common mistakes
Answering cosh(x)+sinh(x)
Adding sinh and cosh produces e^x, which is a different identity. The derivative of sinh is cosh alone.
Copying a trigonometric minus onto sinh(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 cosh(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 sinh(x)?
cosh(x). sinh(x) = (e^x − e^(−x))/2. Differentiating term by term gives (e^x − (−e^(−x)))/2 = (e^x + e^(−x))/2, which is cosh(x). There is no minus in the final answer, unlike cosine. The minus inside the definition is canceled by the derivative of e^(−x).
Which wrong answer shows up for sinh(x)?
Adding sinh and cosh produces e^x, which is a different identity. The derivative of sinh is cosh alone.
Where is the derivative of sinh(x) defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.