Derivative of `sinh(x^2)`

Step-by-step differentiation

Answer

\frac{d}{d x} [sinh (x^{2})]

2 x \cdot cosh (x^{2})

Step-by-step solution

1Chain Rule — sinh
$\frac{d}{d x} [sinh (x^{2})] = cosh (x^{2}) \cdot \frac{d}{d x} [x^{2}]$
The derivative of $sinh (u)$ is $cosh (u)$ , where $u = x^{2}$ . Multiply by $u^{'}$ (chain rule).
↳Find u′: Differentiate the inner function, $u = x^{2}$ :
2Power Rule
$\frac{d}{d x} [x^{2}] = 2 \cdot x^{2 - 1}$
The exponent is $2$ . Bring it down in front as a coefficient, then reduce the power by $1$ .
=Combine
$\frac{d}{d x} [sinh (x^{2})] = cosh (x^{2}) \cdot 2 x = 2 cosh (x^{2}) \cdot x$
With $u^{'} = 2 x$ , multiply by the outer derivative to complete the chain rule.
✓Final Answer
$\frac{d}{d x} [sinh (x^{2})] = 2 x \cdot cosh (x^{2})$
The first derivative simplifies to $2 x \cdot cosh (x^{2})$ .

Understanding this derivative

Differentiating sinh(x²) exercises 3 distinct techniques — the chain rule, the power rule and the hyperbolic derivatives. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

The chain rule handles the composite structure here: differentiate the outer function while leaving the inner expression alone, then multiply by the derivative of that inner expression. Forgetting that final multiplication is one of the most common mistakes in differentiation. The power rule is the workhorse step: bring the exponent down as a coefficient and reduce the exponent by one. Hyperbolic functions mirror their trigonometric cousins but without the sign change: sinh differentiates to cosh and cosh to sinh, both positive.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

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Derivative of sinh(x^2)

Step-by-step solution

Understanding this derivative

Derivative of `sinh(x^2)`