Differentiate the following functions.

1. \( \displaystyle f(x)= \sqrt[7]{x^4}(3x^2+8e^x)\)


Solution


2. \( \displaystyle g(t)=\frac{e^t}{1-2t+3t^2}\)


Solution


3. \( \displaystyle h(u)= \left(\frac{5}{u^2}+2u^3\right)7^u\)


Solution


4. \( \displaystyle f(z)= \frac{az^2}{b+ce^z}\)


Solution


5. Find the equation of the tangent line to the curve \( \displaystyle f(x)= \frac{x}{1-2x^2}\) at \((1,-1)\).


Solution


6. If \(g(x)=xe^x\) find \(g''(1)\).


Solution


7. Suppose that \(f(3)=-1\), \(g(3)=2\), \(f'(3)=-3\), and \(g'(3)= 5\). Find \(h'(3)\).

(a) \(h(x)=2f(x)+8g(x)+2x\)



Solution


(b) \(\displaystyle h(x)= f(x)g(x) \)


Solution


(c) \(\displaystyle h(x)= \frac{f(x)}{g(x)} \)


Solution


(d) \(\displaystyle h(x)=\frac{f(x)+x^2}{g(x)+1}\)


Solution