Question

Derivative using FTC as art 2

Answer 1

`81x\cdot e^(9x)-36x^5\cdot\ln(x)`

Explanation:

You want the derivative ...

`\displaystyle (d)/(dx)\int_(x^6)^{e^(9x)}{ln((t))}\,dt`

Fundamental Theorem of Calculus

The fundamental theorem of calculus tells you the derivative is ...

`\displaystyle (d)/(dx)\int_(u(x))^(v(x)){f(t)}\,dt=f(v(x))v'(x)-f(u(x))u'(x)\\\\\\(d)/(dx)\int_(x^6)^{e^(9x)}{ln((t))}\,dt=\ln(e^(9x))\cdot9e^(9x)-ln((x^6))\cdot6x^5\\\\\\\boxed{81x\cdot e^(9x)-36x^5\cdot\ln(x)}`

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Additional comment

A factor of 9x can be brought out:

= 9x(9e^(9x) -4x^4·ln(x))

You know that ln(e^u) = u, and ln(x^6) = 6·ln(x).

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