Differentiation Questions And Answers Pdf

Differentiating Expressions with Respect to x

Understanding Derivative Rules

Differentiating expressions with respect to x plays a crucial role in calculus. Here's a look at three different expressions and their respective derivatives:

Expression a:

dy/dx (x^4 * 3 * x^2) = 12x^5

Explanation: When differentiating a power multiplied by a constant, multiply the constant by the power and decrement the power by 1.

Expression b:

dy/dx (y * x^2 * x^4) = 6xy^5

Explanation: Treat each factor separately, using the product rule to multiply the derivatives: (y' * x^2 * x^4) + (y * (2x^3) * x^4) = 6xy^5.

Expression c:

dy/dx (2 / x^2 + 1 / 4x) = -4/x^3 - 1/4x^2

Explanation: Apply the quotient rule, which involves dividing the derivative of the numerator by the squared denominator: (0 - 2 * 2x) / x^4 - (1 * 4) / (4x^3) = -4/x^3 - 1/4x^2.