Basic Derivatives using formula

1. Find the derivative, \( f'(x) \), of the function \(f\) using the rules of differentiation. You do not need to simplify your answer.   solution

(a) \( f(x) = x^{2.3} \)   (b) \( f(x) = 120\)   (c) \( f(x) = x^2 -1 \)

(d) \( f(x) = 1+x - x^3 \)   (e) \( f(r) = \pi r^2\)   (f) \( f(r) = \frac{4}{3}\, \pi r^{3} \)

(g) \( f(u) = 5u^2 - \frac{5}{u^2} \)   (h) \( f(x) = 5 \sqrt{x} \)   (i) \( f(t) = -t^2+ t^{-2} \)

(j) \( f(x) = \frac{1}{x} - \frac{3}{\sqrt[3] x } \)

2. Find the derivative of the function \(f\).   solution

(a)   \(\displaystyle f(t)=\frac{2}{7}t^{7/2}\)    (b)   \(\displaystyle f(u)=\frac{5}{\sqrt{u}}-\frac{1}{u\sqrt{u}} \)     (c)   \( \displaystyle f(x)=\frac{x^3-5x^2+7x-2}{x} \)

3. Find the derivative of the function \(f\).  solution

(a) \(f(x)=e^x+x^2-1 \)   (b) \( f(x)=7-\ln x \)   (c) \( f(x)=2x+ 2^x -x^2 \)

4. Let \(f(x)=3x^2-2x\). Find (a) \(f'(1)\),  (b) \(f'(-2)\)  solution

5. Find an equation of the tangent line to \(f(x) = x^3+4\) at the point \((1, 5)\). Write your final answer in the form \(y = mx + b\).  solution