What Is the Gradient? | Gradient & Directional Derivative Explained (Multivariable Calculus)

In this lesson, Professor V explains the gradient as the generalized derivative of a function of several variables. We’ll see why, out of context, it’s just the vector of partial derivatives — but with context it becomes a powerful tool for: ✅ finding the direction of steepest increase ✅ calculating the maximum rate of change ✅ identifying a normal vector to a surface Then we connect the gradient to the directional derivative — in plain language, the rate of change of your function as you move in any chosen direction — using real textbook-style problems: temperature fields, maximum rate of change questions, and equations of planes normal to surfaces. Perfect for students in Multivariable Calculus, Vector Calculus, or anyone reviewing before an exam. 🔗Watch the introductory video lecture on this lesson here:    • Calculus 3: Directional Derivatives and th...   🔗 Watch my full Multivariable Calculus playlist:    • Calculus 3 Video Lectures   xoxo, Professor V 💛 #Gradient #DirectionalDerivative #MultivariableCalculus #VectorCalculus #PartialDerivatives #RateOfChange #MaximumRateOfChange #NormalVector #MathWithProfessorV #CalculusTutorial #MathHelp #studywithme https://linktr.ee/mathwithprofessorv 💡 Support & Exclusive Content 👉 Join my channel for perks:    / @mathwithprofessorv   👉 Become a Patron:   / mathwithprofessorv   Patreon gives you access to exclusive ad-free videos, worksheets, and my full math library. The "Star Pupil Package" tier is only $9.99/month! 📺 Watch Next – Calculus Playlists Full Calculus 3 Video Lectures Playlist:    • Calculus 3 Video Lectures   Full Calculus 1 Video Lectures Playlist:    • Calculus 1 Video Lectures   Full Calculus 2 Integration Playlist:    • Calculus 2 Video Lectures   ✨ Most Helpful Integration Lessons (Calc 2) Integration by parts:    • Calculus 2: Integration by Parts (Video #1...   Tabular Integration:    • Integration by Parts - Tabular Method | Ca...   Trigonometric integrals:    • Calculus 2: Trigonometric Integrals (Video...   Trigonometric Substitution:    • Calculus 2: Trigonometric Substitution (Vi...   Partial Fraction Decomposition:    • Calculus 2: Integration of Rational Functi...   Strategy for Integration:    • Calculus 2: Strategy for Integration (Vide...   Improper Integrals:    • Calculus 2: Improper Integrals (Video #7) ...   📘 Quick Trig Review Before Integrals Unit Circle:    • Trigonometry: Circular Functions (Section ...   Trig Identities:    • Precalculus: Trigonometric Identities (Sec...   Sum & Difference Formulas:    • Precalculus: Sum and Difference Formulas f...   Double & Half-Angle Formulas:    • Precalculus: The Double-Angle and Half-Ang...   🛍️ Shop & Support Merch: https://math-with-professor-v.mysprea... Amazon Influencer Store: https://amazon.com/shop/mathwithprofe... 📲 Stay Connected IG: @mathwithprofessorv TikTok: @mathwithprofessorv This video is copy protected and cannot be downloaded or used in any capacity without my permission. #mathwithprofessorv #integration #partialfractions #trigonometricsubstitution #trigintegrals #integrals #integralcalculus #calculus2 #calculusvideos #integralvideos #trigonometricintegrals #math #youtubemath #mathvideos #mathtutor #mathprofessor #integrationbyparts #calculuspractice #calc2review #integralpractice