Introduction to differential calculus

OVERVIEW

Calculus is a branch of mathematics that deals with the calculation of instantaneous changes (differential calculus) and the summing of an infinite number of small factors to determine some whole (integral calculus). In the 17th century, two mathematicians, Isaac Newton and Gottfried Wilhelm Leibniz, independently developed calculus. Anyone wishing to pursue a degree in physics, chemistry, biology, economics, finance, or actuarial science should start with calculus. With calculus, you can predict a dam's water level or track the position of a space shuttle. Computers have become a valuable tool for solving calculus problems that were once considered impossibly difficult.

Multiple Choice

What is the derivative of xⁿ?

• (n-1)xⁿ

• nxⁿ⁺¹

• (n+1)x^n-1

• nx^n-1

Multiple Choice

Find the derivative and gradient at the pont x=2 $$y=2x^4-3x\$$  

• $$8x^3-3,\ 61$$  

• $$8x^4-3x,\ 138$$  

• $$8x^3-3,\ 67$$  

• $$\frac{2}{5}x^{5\ }-\frac{3}{2}x^2,\ 24$$  

Multiple Choice

Differentiate 6x^⅔.

• 4x^-⅓

• 4x^-⅔

• 6x^⅓

• 4x^⅓

Multiple Choice

The slope of a function is described by its ____________.

• first derivative

• second derivative

• third derivative

• expression

Multiple Choice

Find the derivative of f(x) = 2x⁵+3x

• 10x⁴+3

• 2x⁴+3

• 10x⁵+3

• 10x⁵+3x

Multiple Choice

Find the derivative of:

f(x) = 2x² - 8x + 9

• 4x

• 4x + 8

• 4x - 8

• 9

Multiple Choice

What is the derivative?

$$y=6x^3-3x^2+x-7\$$  

• $$\frac{\text{d}y}{\text{d}x}=18x^2-6x+1$$  

• $$\frac{\text{d}y}{\text{d}x}=18x^3-6x-7$$  

• $$\frac{\text{d}y}{\text{d}x}=6x^2-3x+1$$  

• $$\frac{\text{d}y}{\text{d}x}=18x^2-6x+1-7$$  

Multiple Choice

Find the derivative of f(x)=(x³-2x)²

• 6x⁵ - 12x³+8x
• 6x⁵ - 16x³+8x
• x⁶-4x⁴+4x²
• 6x⁵ - 16x^3-8x

Multiple Choice

Find dy/dx for y=-(3x²+5x)⁵

• y=-5(x+5)⁴
• y=-5(6x+5)(3x²+5x)⁴
• y=-6x+5(3x²+5x)⁴
• y=-6x(3x²+5x)⁴

Multiple Choice

Find the derivative of:
f(x) = 7(3x + 4)⁵

• 35(3x + 4)⁵
• 35(3x + 4)⁴
• 105(3x + 4)⁵
• 105(3x + 4)⁴