Find out the answer to "Calculate the derivative. ddx∫x7e9xln⁡(t)dt"

Lennie Carroll

Answered question

2021-10-29

Calculate the derivative.

\frac{d}{d x} \int_{x^{7}}^{e^{9 x}} \ln (t) d t

Clelioo

Skilled2021-10-30Added 88 answers

Step 1
First, we split the integral function into two terms.

\int_{x^{7}}^{e^{9 x}} \ln (t) d t = \int_{x^{7}}^{c} \ln (t) d t + \int_{c}^{e^{9 x}} \ln (t) d t = \int_{c}^{e^{9 x}} \ln (t) d t - \int_{c}^{x^{7}} \ln (t) d t

Now find the derivative of both integrals by using the chain rule

\frac{d}{d x} \int_{c}^{e^{9 x}} \ln (t) d t

= \ln (e^{9 x}) \cdot {(e^{9 x})}^{p r i m e}

= 9 x \cdot 9 e^{9 x}

= 81 x e^{9 x}

Step 2
and

\frac{d}{d x} \int_{c}^{x^{7}} \ln (t) d t

= \ln (x^{7}) \cdot {(x^{7})}^{p r i m e}

= \ln (x^{7}) \cdot 7 x^{6}

= 7 x^{6} \ln (x^{7})

By substituting these two integrals in above.
Thus, the derivative of the integral function is

= 81 x e^{9 x} - 7 x^{6} \ln (x^{7})

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