Derivatives - Excercise 1

The derivative measures how a function changes with respect to its variable. It reveals slopes of tangents, extrema (maxima and minima), and inflection points. Below you will find a concise recap of core ideas and a step‑by‑step solution of three practice derivatives that focus on the power rule.

Basic concepts

The derivative of a function f(x) at a point x can be defined by the limit of the average rate of change as the increment h tends to 0:

f'(x) = lim_{h->0} ( f(x + h) - f(x) ) / h

This definition underpins all differentiation rules used in practice.

Rules for differentiation

• derivative of a constant: d/dx(c) = 0
• derivative of a power: d/dx(x^n) = n*x^(n-1)
• constant multiple rule: d/dx(kx^n) = kn*x^(n-1)
• sum and difference: differentiate term by term
• product rule: d/dx( f(x)*g(x) ) = f'(x)*g(x) + f(x)*g'(x)
• quotient rule: d/dx( f(x)/g(x) ) = ( f'(x)*g(x) - f(x)*g'(x) ) / ( g(x) )^2
• chain rule (composition): d/dx( f(g(x)) ) = f'(g(x))*g'(x)

In the worked examples below we mainly use the power rule and linearity (sum and constant multiple rules).

Examples

Example 1
Given f(x) = x^-3. Use the power rule with n = -3.
f'(x) = -3x^(-3-1) = -3x^-4
(Optional rewrite) f'(x) = -3/(x^4), with x != 0.

Example 2
Given f(x) = 3x^8 + x^-2. Differentiate term by term.
d/dx(3x^8) = 38x^(8-1) = 24x^7
d/dx(x^-2) = -2x^(-2-1) = -2x^-3
Combine: f'(x) = 24x^7 - 2x^-3
(Optional rewrite) f'(x) = 24x^7 - 2/(x^3), x != 0.

Example 3
Given f(x) = 3x^4 - 5x^3 + 2x^2 + 6x + 8. Differentiate each term.
d/dx(3x^4) = 12x^3
d/dx(-5x^3) = -15x^2
d/dx(2x^2) = 4x
d/dx(6x) = 6
d/dx(8) = 0
Combine: f'(x) = 12x^3 - 15x^2 + 4x + 6

Uses of derivatives

Derivatives help you find instantaneous rates of change, optimize functions (maxima and minima), analyze concavity and inflection points, and build linear approximations. They also form the foundation for integrals, differential equations, and many applications in physics, engineering, and data science.

Conclusion

Mastery comes from repetition. Apply the power rule carefully, reduce the exponent by 1 after bringing it down, track signs with negative exponents, and remember that d/dx(x) = 1 and d/dx(constant) = 0. With these habits, the three examples above become routine—and so will much harder ones.