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 the graph of f(x) = 1/x, identify its two branches, and grasp limits as x approaches infinity and zero from both sides to prepare for differentiation.

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.

Learn to evaluate limits by substitution, factoring, and cancellation, handle undefined points, and analyze left- and right-hand approaches and limits at infinity.

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.

Learn to find the first derivative of polynomials with the limit method, deriving the gradient function dy/dx. See that y = 2x^2 yields dy/dx = 4x at any x.

Apply the limit method to find the first derivative of polynomials by forming delta y over delta x and taking the limit as delta x approaches zero, introducing dy/dx.

Derive derivatives from first principles using the limit definition and compute dy/dx for functions shown in the quiz. Analyze instantaneous rate of change and gradient as delta x approaches zero.

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.

Derive the power rule for differentiation and see how the derivative of x^n equals n x^{n-1}, a fundamental formula for future exercises.

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.

Apply the prime differentiation formula to polynomials and powers, compute dy/dx at x=1, and work through quiz-style differentiation problems.

Learn the product rule for differentiation, including the formula dy/dx = u'v + uv', derive it via limits, and apply it to examples like (2x+1)(3x-5) to compare methods.

Learn to differentiate using the product rule, applying u'v plus uv', solving multiple examples, and computing derivatives and instantaneous gradients at given x-values.

Master the quotient rule for differentiating a ratio of functions. Apply the formula (v u' - u v')/v^2 and practice with examples like (2-3x)/(3x+1) to sharpen understanding.

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.

Apply the chain rule to differentiate composite functions by using an inner function u and dy/dx = dy/du × du/dx. Practice with examples like (x-3)^7 and other inner–outer configurations.

Learn to differentiate using the chain rule on embedded functions, identify outer and inner components, and solve five quiz problems with dy/dx calculations.

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.

Learn to differentiate implicit functions using implicit differentiation, applying chain and product rules to find dy/dx and gradients, then evaluate at specified points for quiz problems.

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

Practice solving differentiation problems on polynomial functions up to the second derivative, using product, quotient, and chain rules, with step-by-step explanations and factorization techniques.

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.

Identify turning points, stationary points, maximum and minimum points, and inflection points on curves using differentiation, where dy/dx equals zero for maxima and minima, and curvature changes at inflection points.

Locate turning points of polynomial functions by solving f'(x)=0. Classify as maxima or minima by inspecting left and right gradients and the leading coefficient for even-degree curves.

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.

Apply the second derivative test to classify turning points as minima or maxima on cubic and other polynomial curves, using implicit differentiation and exploring inflection points.

This lecture covers solving quiz questions to find maximum and minimum points using the second derivative test, locating turning points and classifying them as minima or maxima.

Determine a and b for f(x)=a x^4 + b x^3 + 5 to yield a minimum at x = -1. Use f'(x)=0 and f''(x)=0 to locate stationary and inflection points.

Explore how to solve geometric optimization problems with differentiation, using a wire-to-cube example to maximize volume while meeting a fixed length, and identify turning points and maximum values.

Explore maximizing a cylinder's volume from a fixed aluminum sheet by expressing volume in terms of radius under the given surface area, and identify the optimal radius and height.

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.

Relate the rate of volume change to the rate of height change in a conical cup. Use v = (1/27) pi h^3 and r = h/3 to find dh/dt when h is 3 cm.

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.

Solve quiz problems on rates of change using the chain rule to relate volume, height, and radius for cylinders and spheres, and compute dV/dt, dh/dt, and dr/dt from given data.

Apply the first derivative to approximate small changes in y from slight x changes. See y = x^3 + 3 and a cone volume example comparing approximate versus actual change.

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.

Explore the core ideas of differentiation in differential calculus, focusing on its role as the instantaneous rate of change and its connection to integration.