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1. Write the product function
![{\displaystyle f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074)
and its rate of change function for
![{\displaystyle g(x)=2e^{5}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a8f292fc1f2d68e7c1296f742c2d2a44690a63)
and
![{\displaystyle h(x)=2x^{2}-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d5b8f3189f732e9b2cabd411cb1da503d05a42)
.
![{\displaystyle f(x)=2e^{5}x(2x^{2}-x),\ f'(x)=2e^{5}(2x^{2}-x)+2e^{5}x(4x-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/520a79ca7a8d087596daf90556cf63b2ac410842)
![{\displaystyle f(x)=2e^{5}x(2x^{2}-x),\ f'(x)=2e^{5}(2x^{2}-x)+2e^{5}x(4x-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/520a79ca7a8d087596daf90556cf63b2ac410842)
2. Write the product function
![{\displaystyle f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074)
and its rate of change function for
![{\displaystyle g(x)=0.5\cos(4x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d862aa68e9a8f90fc93b37112143f115fc8910)
and
![{\displaystyle h(x)=2\ln(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c24033c0a0e480f5e11d4aad8b2890fdd7de3d04)
.
![{\displaystyle f(x)=\cos(4x)\ln(x),\ f'(x)=-4\sin(4x)\ln(x)+{\frac {\cos(4x)}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45d4b63ca93f35ba573f6514e3aa9858be572042)
![{\displaystyle f(x)=\cos(4x)\ln(x),\ f'(x)=-4\sin(4x)\ln(x)+{\frac {\cos(4x)}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45d4b63ca93f35ba573f6514e3aa9858be572042)
3. Find
![{\displaystyle f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd7d7c75340e779d82658e19d1720ce84ab127)
if
![{\displaystyle f(x)=(2x+3)e^{-x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5e5bcedb3ff995e82fea987980ef6dd0623857)
.
![{\displaystyle 2e^{-x}-(2x+3)e^{-x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0262e46631f3b9f81ed6c6823d43fa96a621574c)
![{\displaystyle 2e^{-x}-(2x+3)e^{-x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0262e46631f3b9f81ed6c6823d43fa96a621574c)
4. Find
![{\displaystyle f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd7d7c75340e779d82658e19d1720ce84ab127)
if
![{\displaystyle f(x)=2{\sqrt {x^{3}}}\ln(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd46dcdf649e0532a1560d96a45c720f72d4036)
.
![{\displaystyle 3{\sqrt {x}}\ln(x)+2{\sqrt {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df0bac826bb8c03e2359a5813a9509e96e53a980)
![{\displaystyle 3{\sqrt {x}}\ln(x)+2{\sqrt {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df0bac826bb8c03e2359a5813a9509e96e53a980)
5. Find
![{\displaystyle f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd7d7c75340e779d82658e19d1720ce84ab127)
if
![{\displaystyle f(x)=(x+2)^{3}(1-x^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/811f40b0899a1d9b74ee4311c9573f742ff3ec1c)
.
![{\displaystyle 3(x+2)^{2}(1-x^{2})-2x(x+2)^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4aef4132dd4bc34e36d29fdeac6ea8728015fa)
![{\displaystyle 3(x+2)^{2}(1-x^{2})-2x(x+2)^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4aef4132dd4bc34e36d29fdeac6ea8728015fa)
6. Find
![{\displaystyle f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd7d7c75340e779d82658e19d1720ce84ab127)
if
![{\displaystyle \displaystyle f(x)={\frac {e^{x}}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2823fdd2ed06e3c59c01a84209ab712b22937a0)
.
![{\displaystyle \displaystyle {\frac {xe^{x}-e^{x}}{x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d05db9853fa832ec4df332073bc73627f20a2046)
![{\displaystyle \displaystyle {\frac {xe^{x}-e^{x}}{x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d05db9853fa832ec4df332073bc73627f20a2046)