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Full access to solution steps Full access to AI chat Practice and improve Access from any device Unlimited storage Detect AI-generated content Advanced Grammar Checker Paraphrase in unlimited modes Summarize any text Back to School Promotion qb-banner-title 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 First Derivative 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 Slope of Tangent Curved Line Slope 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 Multi Variable Limit Specify Method L'Hopital's Rule Squeeze Theorem Chain Rule Factoring Substitution Sandwich Theorem L'Hopital's Rule Squeeze Theorem 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 Indefinite Integrals Definite Integrals Specific-Method Partial Fractions U-Substitution Trigonometric Substitution Weierstrass Substitution By Parts Long Division Partial Fractions Trigonometric Substitution Weierstrass Substitution 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 Area under curve Area between curves Area under polar curve Volume of solid of revolution 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 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 Geometric Series Test Telescoping Series Test Alternating Series Test Divergence Test Comparison Test Limit Comparison Test Absolute Convergence Power Series Radius of Convergence Interval of Convergence 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 Linear First Order Linear w/constant coefficients IVP using Laplace Series Solutions Method of Frobenius Multivariable Calculus Partial Derivative Implicit Derivative Tangent to Conic Multi Variable Limit Multiple Integrals Gradient Divergence Extreme Points Partial Derivative Implicit Derivative Tangent to Conic Multi Variable Limit Multiple Integrals Laplace Transform Inverse Taylor/Maclaurin Series Taylor Series Maclaurin Series Maclaurin Series Fourier Transform \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)) 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

$\frac{d}{dx}\left(x^n\right)=nx^{n-1}$ 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 : If $f\left(x\right)=5$ , then, $f'\left(x\right)=0$ Constant Multiple Rule : $\frac{\mathrm{d} (cf(x))}{\mathrm{d} x} = c\frac{\mathrm{d} (f(x))}{\mathrm{d} x}$ Example 3 : If $f\left(x\right)=4x^7$, then, $f'\left(x\right)=4\times 7x^6$ $f'\left(x\right)=28x^6$ $\frac{\mathrm{d} (f(x)+g(x))}{\mathrm{d} x} = f'(x)+g'(x)$ Example 4 : If $f\left(x\right)=x^3+2x^2+7$, then, $f'\left(x\right)=3x^2+4x+0$ Quotient Rule : $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^{2}}$ Example 5 : If $f\left(x\right)=3x+9$ and $g\left(x\right)=2-x$, then find $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)$. $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right) = \frac{d}{dx}\left(\frac{3x+9}{2-x}\right)$ Applying quotient rule $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^{2}}$ $\frac{d}{dx}\left(\frac{3x+9}{2-x}\right) = \frac{\frac{d}{dx}\left(3x+9\right)\left(2-x\right)-\frac{d}{dx}\left(2-x\right)\left(3x+9\right)}{\left(2-x\right)^2}$ As $\frac{d}{dx}\left(3x+9\right)=3$ and $\frac{d}{dx}\left(2-x\right)=-1$, $\frac{d}{dx}\left(\frac{3x+9}{2-x}\right) = \frac{3\left(2-x\right)-\left(-1\right)\left(3x+9\right)}{\left(2-x\right)^2}$ $=\frac{15}{\left(2-x\right)^2}$ So, $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{d}{dx}\left(\frac{3x+9}{2-x}\right) = \frac{15}{\left(2-x\right)^2}$ $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=f'g(x)\cdot g'(x)$ Example 6 : If $f\left(x\right)=x^2$ and $g\left(x\right)=2x+1$, find $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}.$ $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=f'g(x)\cdot g'(x)$ Now, $f'(x)=2x$ and $g'(x)=2$ $f'\left(g\left(x\right)\right)=\text{f}'\left(2x+1\right)$ $\text{f}'\left(2x+1\right)=2\left(2x+1\right)=4x+2$ $f'\left(g\left(x\right)\right)\cdot \text{g}'\left(x\right)=2\left(4x+2\right)=8x+4$ So, $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=8x+4$ $\frac{d}{dx}\left(f\left(x\right)\cdot g\left(x\right)\right)=f(x)\cdot g'(x)+f'(x)\cdot g(x)$ Common Derivative Formulas $\frac{d}{dx}\left(e^x\right) = e^x$ $\frac{d}{dx}\left(e^x\right) = e^x$ $\frac{d}{dx}\left(\ln \left(x\right)\right) = \frac{d}{dx}\left(\ln \left(x\right)\right)$ $\frac{d}{dx}\left(\ln \left(x\right)\right) = \frac{d}{dx}\left(\ln \left(x\right)\right)$ $\frac{d}{dx}\left(\sin \left(x\right)\right) = \cos \left(x\right)$ $\frac{d}{dx}\left(\sin \left(x\right)\right) = \cos \left(x\right)$ $\frac{d}{dx}\left(\cos \left(x\right)\right) = -\sin \left(x\right)$ $\frac{d}{dx}\left(\cos \left(x\right)\right) = -\sin \left(x\right)$ $\frac{d}{dx}\left(\tan \left(x\right)\right) = \sec ^2\left(x\right)$ $\frac{d}{dx}\left(\tan \left(x\right)\right) = \sec ^2\left(x\right)$ $\frac{d}{dx}\left(\sec \left(x\right)\right) = \sec \left(x\right)\tan \left(x\right)$ $\frac{d}{dx}\left(\sec \left(x\right)\right) = \sec \left(x\right)\tan \left(x\right)$ $\frac{d}{dx}\left(\cosec \left(x\right)\right) = -\cot \left(x\right)\cosec \left(x\right)$ $\frac{d}{dx}\left(\cosec \left(x\right)\right) = -\cot \left(x\right)\cosec \left(x\right)$ $\frac{d}{dx}\left(\cot \left(x\right)\right) = -\cosec ^2\left(x\right)$ $\frac{d}{dx}\left(\cot \left(x\right)\right) = -\cosec ^2\left(x\right)$ Example : Find the derivative of $f\left(x\right)=\frac{1}{x}$. Solution : We can rewrite $\frac{1}{x}$ as $f'\left(x\right)=\left(-1\right)x^{-1-1}$ $f'\left(x\right) = -x^{-2}$ Example : Find $\frac{d}{dx}\left(\sin \left(x\right)\cdot \text{e}^x\right)$. Solution : Using product rule, $\frac{d}{dx}\left(f\left(x\right)\cdot g\left(x\right)\right)=f(x)\cdot g'(x)+f'(x)\cdot g(x)$ Here, $f\left(x\right)=\sin \left(x\right)$ and $g\left(x\right)=e^x$ $f'\left(x\right)=\cos \left(x\right)$ and $g'\left(x\right)=e^x$ So, $\frac{d}{dx}\left(\sin \left(x\right)\cdot \text{e}^x\right)=\left(cos\left(x\right)\right)\cdot e^x+\left(\sin \left(x\right)\right)\cdot e^x$ Example : Differentate $y=\ln\left(x^2+1\right)$. Here, we would use chain rule. $f\left(g\left(x\right)\right)=\ln \left(g\left(x\right)\right)$ and $\text{g}\left(x\right)=\text{x}^2+1$ So, $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=f'g(x)\cdot g'(x)$ $\text{f}'\left(g\left(x\right)\right)=\frac{1}{\text{x}^2+1}$ and $g'\left(x\right) = 2x$ $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=\frac{2x}{\text{x}^2+1}$ Example : Find the derivative of $y=\frac{x^2+1}{x}$. Solution : Here, we would use the quotient rule. $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^{2}}$ $f\left(x\right)=x^2+1$ and $g\left(x\right)=x$ $f'\left(x\right)=2x$ and $g'\left(x\right)=1$ $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{\left(2x\cdot x\right)-\left(x^2+1\right)}{x^2}$ = $\frac{2x^2-x^2-1}{x^2}$ $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{x^2-1}{x^2}$ Example : Differentiate $f\left(x\right)=\sin\left(x^2\right)\cdot \cos\left(x\right)$ Solution : Here, we would use both chain rule and product rule. Let $u=sin\left(x^2\right)$ and $v=\cos\left(x\right)$ $u'=\cos^{ }\left(x^2\right)\cdot 2x$ and $v'=-sin\left(x\right)$ $f'\left(x\right)=\text{u}'v+\text{uv}'$ $f'\left(x\right)=2xcos\left(x^2\right)\cdot cos\left(x\right)-\sin\left(x^2\right)\cdot sin\left(x\right)$ Real-Life Applications of Derivatives Physics : Derivatives are used to determine velocity (rate of change of position) and acceleration (rate of change of velocity). Physics : Derivatives are used to determine velocity (rate of change of position) and acceleration (rate of change of velocity). Economics : Derivatives help calculate marginal cost and marginal revenue, essential in optimizing production and profits. Economics : Derivatives help calculate marginal cost and marginal revenue, essential in optimizing production and profits. Biology : The growth rates of populations are modeled through derivatives. Biology : The growth rates of populations are modeled through derivatives. Engineering : Derivatives are used in analysing velocity, acceleration, jerk etc. and modeling systems that change over time. Engineering : Derivatives are used in analysing velocity, acceleration, jerk etc. and modeling systems that change over time. How to use a Limits Calculator Enter Your Problem: Type in your equation , expression, or system into the calculator's input field. Select the operation : Choose the function you need: solve, simplify, factor, graph, etc. Click Calculate : The calculator processes your input and provides a detailed solution. Review the Steps : The step-by-step explanation helps you understand the process and learn how to solve similar problems. Solve for f'(x) if f(x) = $\frac{x^2+3}{x}$ Step 1 : Open the calculator. Step 2 : Select the $\frac{d}{dx}$ option. Step 3 : Now choose the fraction option. Step 4 : Write $x^2+3$ in its numertor and x in its denominator. Step 5 : Press ‘Go’ and you can see the step-wise solution there. Benefits of Using Derivative Calculator Saves time and provides accurate solutions. Saves time and provides accurate solutions. Shows step-by-step solutions for learning. Shows step-by-step solutions for learning. Useful for students and teachers. Useful for students and teachers. Online accessibility and free usage. Online accessibility and free usage. How do you calculate derivatives? To calculate derivatives start by identifying the different components (i.e. multipliers and divisors), derive each component separately, carefully set the rule formula, and simplify. If you are dealing with compound functions, use the chain rule. Is there a calculator for derivatives? Symbolab is the best derivative calculator, solving first derivatives, second derivatives, higher order derivatives, derivative at a point, partial derivatives, implicit derivatives, derivatives using definition, and more. Is velocity the first or second derivative? Velocity is the first derivative of the position function. Acceleration is the second derivative of the position function. What is the derivative of a Function? The derivative of a function represents its a rate of change (or the slope at a point on the graph). What is the derivative of zero? The derivative of a constant is equal to zero, hence the derivative of zero is zero. What does the third derivative tell you? The third derivative is the rate at which the second derivative is changing. derivative-calculator Advanced Math Solutions – Derivative Calculator, Implicit Differentiation We’ve covered methods and rules to differentiate functions of the form y=f(x), where y is explicitly defined as... Please add a message. Message received. Thanks for the feedback.