Question

Please find the derivative h(x)= sinh(x^5)

Answer 1

To find the derivative of h(x) = sinh(x^5), we use the chain rule, resulting in the derivative being 5x^4 × cosh(x^5).

Step-by-step explanation:

The question involves finding the derivative of the function h(x) = sinh(x^5). To find the derivative, we need to apply the chain rule. The derivative of the hyperbolic sine function, sinh(u), with respect to u, is cosh(u). When u = x^5, we differentiate x^5 with respect to x to get 5x^4. Therefore, the derivative of h(x) with respect to x is 5x^4 × cosh(x^5).

Answer 2

Recall that:

`(d)/(du)\sinh (u)=\cosh (u).`

Applying the chain rule we get that:

`(d)/(dx)\sinh (x^5)=(d)/(du)\sinh (u)|_(u=x^5)\cdot(d)/(dx)(x^5)\text{.}`

Therefore:

`(d)/(dx)\sinh (x^5)=\cosh (u)|_(u=x^5)\cdot5x^(5-1)\text{.}`

Simplifying the above result we get:

`\begin{gathered} (d)/(dx)\sinh (x^5)=\cosh (x^5)\cdot5x^4 \\ =5x^4\cosh (x^5)\text{.} \end{gathered}`

Answer:

`h^(\prime)(x)=5x^4\cosh (x^5)\text{.}`