Differentiate the following functions.

1. \( \displaystyle f(x)=\pi^7\)


Solution


2. \(\displaystyle F(x)=\frac{2}{3} x^{12}- \sqrt{x} + e^{19}\)


Solution


3. \(\displaystyle g(u)= \frac{2}{u^5}-\frac{u^6}{3}+3 \sqrt[3]{u^2}-\ln 2\)


Solution


4. \(\displaystyle h(x)=2x^5(x^3-4x^2+4) \)


Solution


5. \(\displaystyle f(t) = (t-5)^2 \)


Solution


6. \(\displaystyle f(x)= e^x+ \sqrt[7]{x^2} + \pi ^3\)


Solution


7. \(\displaystyle A(y)= cy^4-2by\)


Solution


8. \(\displaystyle V(r)= \frac{4}{3} \pi r^3\)


Solution


9. \(\displaystyle f(x)= \frac{2x^4-3x^3+2x^2}{x^3}\)


Solution


10. \(\displaystyle f(x) = e^x+x^e\)


Solution


11. \(\displaystyle f(x) = \sqrt{5 x} -\sqrt{5}{x}\)


Solution


12. The equation of motion of a particle is \(s(t)=2t^3-4t^2+5t-30\), where \( s\) is in meters and \(t\) is in seconds.

(a) Find the velocity and acceleration functions as functions of \(t\).

(b) Find the acceleration after 3 s.

(c) Find the velocity after 3 s.


Solution


13. Find the points on the curve \(\displaystyle y= \frac{1}{4}x^4+\frac{1}{3}x^3-x^2+3\) where the tangent is horizontal.


Solution