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Practice More Type your Answer Verify x^2 x^{\msquare} \log_{\msquare} \sqrt{\square} \nthroot[\msquare]{\square} \le \ge \frac{\msquare}{\msquare} \cdot \div x^{\circ} \pi \left(\square\right)^{'} \frac{d}{dx} \frac{\partial}{\partial x} \int \int_{\msquare}^{\msquare} \lim \sum \infty \theta (f\:\circ\:g) f(x) ▭\:\longdivision{▭} \times \twostack{▭}{▭} + \twostack{▭}{▭} - \twostack{▭}{▭} \left( \right) \times \square\frac{\square}{\square} Take a challenge Subscribe to verify your answer Subscribe Are you sure you want to leave this Challenge? 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Go back Purchase Bundle Back to School Promotion Annual Annual - $ % OFF Annual plan One time offer for one year, then $ Go back + qb-banner-title Solutions > Calculus Calculator > Derivative Calculator Topic Pre Algebra Algebra Pre Calculus Calculus Derivatives First Derivative WRT Specify Method Chain Rule Product Rule Quotient Rule Sum/Diff Rule Second Derivative Third Derivative Higher Order Derivatives Derivative at a point Partial Derivative Implicit Derivative Second Implicit Derivative Derivative using Definition Derivative Applications Tangent Slope of Tangent Normal Curved Line Slope Extreme Points Tangent to Conic Linear Approximation Difference Quotient Horizontal Tangent Limits One Variable Multi Variable Limit One Sided At Infinity Specify Method L'Hopital's Rule Squeeze Theorem Chain Rule Factoring Substitution Sandwich Theorem Integrals Indefinite Integrals Definite Integrals Specific-Method Partial Fractions U-Substitution Trigonometric Substitution Weierstrass Substitution By Parts Long Division Improper Integrals Antiderivatives Double Integrals Triple Integrals Multiple Integrals Integral Applications Limit of Sum Area under curve Area between curves Area under polar curve Volume of solid of revolution Arc Length Function Average Integral Approximation Riemann Sum Trapezoidal Simpson's Rule Midpoint Rule Series Convergence Geometric Series Test Telescoping Series Test Alternating Series Test P Series Test Divergence Test Ratio Test Root Test Comparison Test Limit Comparison Test Integral Test Absolute Convergence Power Series Radius of Convergence Interval of Convergence ODE Linear First Order Linear w/constant coefficients Separable Bernoulli Exact Second Order Homogenous Non Homogenous Substitution System of ODEs IVP using Laplace Series Solutions Method of Frobenius Gamma Function Multivariable Calculus Partial Derivative Implicit Derivative Tangent to Conic Multi Variable Limit Multiple Integrals Gradient Divergence Extreme Points Laplace Transform Inverse Taylor/Maclaurin Series Taylor Series Maclaurin Series Fourier Series Fourier Transform Functions Linear Algebra Trigonometry Statistics Physics Chemistry Finance Economics Conversions Add to Chrome Get our extension, you can capture any math problem from any website Full pad x^2 x^{\msquare} \log_{\msquare} \sqrt{\square} \nthroot[\msquare]{\square} \le \ge \frac{\msquare}{\msquare} \cdot \div x^{\circ} \pi \left(\square\right)^{'} \frac{d}{dx} \frac{\partial}{\partial x} \int \int_{\msquare}^{\msquare} \lim \sum \infty \theta (f\:\circ\:g) f(x) - \twostack{▭}{▭} \lt 7 8 9 \div AC + \twostack{▭}{▭} \gt 4 5 6 \times \square\frac{\square}{\square} \times \twostack{▭}{▭} \left( 1 2 3 - x ▭\:\longdivision{▭} \right) . 0 = + y \mathrm{implicit\:derivative} \mathrm{tangent} \mathrm{volume} \mathrm{laplace} \mathrm{fourier} See All area asymptotes critical points derivative domain eigenvalues eigenvectors expand extreme points factor implicit derivative inflection points intercepts inverse laplace inverse laplace partial fractions range slope simplify solve for tangent taylor vertex geometric test alternating test telescoping test pseries test root test Go Steps Graph Related Examples Generated by AI AI explanations are generated using OpenAI technology. AI generated content may present inaccurate or offensive content that does not represent Symbolab's view. Verify your Answer Subscribe to verify your answer Subscribe Save to Notebook! Sign in to save notes Sign in Verify Save Show Steps Hide Steps Number Line Related Derivative Examples \frac{d}{dx}(\frac{3x+9}{2-x}) \frac{d^2}{dx^2}(\frac{3x+9}{2-x}) (\sin^2(\theta))'' derivative\:of\:f(x)=3-4x^2,\:\:x=5 implicit\:derivative\:\frac{dy}{dx},\:(x-y)^2=x+y-1 \frac{\partial}{\partial y\partial x}(\sin (x^2y^2)) \frac{\partial }{\partial x}(\sin (x^2y^2)) Show More Derivative Calculator – Step by Step Guide to Solving Derivatives Online Imagine travelling in a car. One hour has passed and you see that you have travelled 30 miles. So, your average speed is 30 miles/hour. But what if someone asks what your speed was at the 20 minute mark, or at the 35 minute mark was? You were not moving with 30 miles/hour speed the whole time, right? This is where derivative comes into play. Whether we're studying the motion of planets, optimizing resources in economics, or analyzing how fast or how slow a car is moving, derivatives are the mathematical lens through which we understand change itself. A brief history The concept of change, the base of derivatives, has intrigued mankind for centuries. The foundation of such concept appears in ancient Greek mathematics, where scientists like Archimedes learnt about change, motion, tangent etc. laying groundwork for later ideas of derivatives. Although the formal concept of derivatives came in the 17th century when calculus was birthed, two scientists, Issac Newton from England and Gottfried Wilhelm Leibniz from Germany, individually developed the core ideas of calculus around the same time. Newton was intrigued by how objects moved, how their positions changed with respect to time, leading him to define what we now call velocity and acceleration using early derivative concepts. Leibniz, alternatively, focused on notation and structure. His elegant notation for derivatives, like $\frac{dy}{dx}$ is widely used till date. Basic concept and definition At the core level, derivative tells us how any quantity is changing with respect to another quantity at an exact point. Mathematically, it is defined as: $f'\left(x\right)=\lim _{h\to 0}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)$ This expression is called first principle of derivatives and it tells us about the change in a function's output when input is changed by a very small amount. Geometrical Interpretation Geometrically , derivative at a point is the slope of the tangent to a curve at that point. If that slope is positive, the quantity is increasing, if it is negative, the quantity is decreasing. Common Derivative Rules Power Rule : $\frac{d}{dx}\left(x^n\right)=nx^{n-1}$ Example 1 : If $f\left(x\right)=x^5$, then, $f'\left(x\right)=5x^4$ Constant Rule : $\frac{d}{dx}\left(c\right)$ = 0 Example 2