Introduction to Differentiation (Differential Calculus)

Explore how limits underpin differentiation by deriving the differentiation formula and examining f(x)=(x^2-1)/(x-1) as x approaches 1, highlighting undefined values and the limit concept.

Explore how limits describe a function’s behavior as x approaches infinity or minus infinity, using terms and neglecting constants; see examples that yield 0 and 1/2 and relate to differentiation.

Explore how differentiation measures rate of change and the instantaneous gradient at a point, using the gradient of a chord on a curve and the tangent line with limits.

Estimate the instantaneous gradient of y = 2x - 1 by using a chord between nearby points and delta y over delta x toward the tangent.

Explore deriving the differentiation formula using the limit method on polynomials, proving dy/dx = n a x^{n-1} for y = a x^n, through structured polynomial examples.

Learn to apply the prime differentiation formula to exam-style problems by differentiating each term of polynomials and using addition rules, and rewrite quotients to find the first derivative.

Learn how to compute the first derivative using the differentiation formula, form the gradient function, and determine tangent line slopes at given points using the constant multiple rule.

Learn how to differentiate quotients using the quotient rule, applying u and v, and compute and verify derivatives at specific x-values through a five-question quiz.

Learn to differentiate implicit functions by converting to explicit form or differentiating implicitly, using circle examples like x^2+y^2=25 to find dy/dx at (4,3).

Learn how to differentiate implicit functions when y is not explicit in x, using product and chain rules to find dy/dx with practical examples.

Explore higher order differentiation from first to fourth derivatives, compare notations, and compute successive derivatives of polynomials, noting when they reach zero.

Learn to differentiate polynomial functions and compute the second derivative using quotient and chain rules, then evaluate f''(3) from the given expression.

Learn to find tangent and normal line equations for a curve at a given point by differentiating the curve and applying the slope product equals -1.

Find tangent and normal lines to a parabola by differentiation to obtain the gradient, use the line x+3y=10 to set perpendicular slopes, locate the point, and write the equations.

Explore derivative and gradient calculations for a polynomial curve to determine tangent and normal lines; apply dy/dx, gradient formulas, and point-slope equations at specific points.

Use the second derivative at turning points where the first derivative is zero to classify maxima and minima by the sign of the second derivative.

Use differential calculus to maximize area under a fixed wire length by expressing the area in x and y, applying the perimeter constraint, and using first and second derivative tests.

Demonstrate differentiation as both a gradient and rate of change, using f(x)=x^2-2x+1 to show the derivative and tangent. Apply the chain rule to relate dA/dt to dr/dt in A=pi r^2.

By using similar triangles, relate radius to height and express the cone’s volume as V=(π/27)h^3. Differentiating with respect to time yields dV/dt=(π/9)h^2 dh/dt; at h=3 and dh/dt=1, dV/dt=π cm^3/s.

Apply differential calculus to approximate small changes in volume for a cube, a sphere, and a cylinder, using derivatives and delta changes to estimate outcomes.