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Chain Rule The chain rule is used to find the derivatives of composite functions like (x 2 1) 3, (sin 2x), (ln 5x), e 2x, and so on If y = f(g(x)), then y' = f'(g(x)) g'(x) The chain rule states that the instantaneous rate of change of f relative to g relative to x helps us calculate the instantaneous rate of change of f relative to x From this formula, we also know that the derivative of a constant is 0 So, the derivative of 3 is 0 and the derivative of 4 is also 0 2 Derivative Of f(x) = sin x And f(x) = cos xInteractive graphs/plots help visualize and better understand the functions

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How to find derivatives of e

How to find derivatives of e-The derivative of ex is ex General Exponential Function a x Assuming the formula for e ;General Derivative Formulas 1) d d x ( c) = 0 where c is any constant 2) d d x x n = n x n – 1 is called the Power Rule of Derivatives 4) d d x f ( x) n = n f ( x) n – 1 d d x f ( x) is the Power Rule for Functions 9) d d x f ( x) ⋅ g ( x) = f ( x) d d x g ( x) g

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Derivative Calculator Step 1 Enter the function you want to find the derivative of in the editor The Derivative Calculator supports solving first, second, fourth derivatives, as well as implicit differentiation and finding the zeros/roots You can also get a better visual and understanding of the function by using our graphing toolThis formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus Derivatives to n th order edit Some rules exist for computing the n th derivative of functions, where n is a positive integer Examples of the first derivative of some worksheet formulas calculated by the custom function are shown in Figure 618 The formula in cell D3 is = FirstDerivDemo (C3,) The formulas labeled "exact" in column E are the appropriate formulas from differential calculus for the first derivative of the respective functions

 Euler's formula is an identity it's true for all x, so it's not possible to solve for x Added We know that ##\frac d {dx} e^x = e^x##, and you have found that ##\frac d {dx} e^{ix} = e^{ix}## as well Is that so surprising?The derivative of 1f = −f'f 2 With f(x)= x, we know that f'(x) = 1 So the derivative of 1x = −1x 2 Which is the same result we got above using the Power RuleUse the formula ex h = exeh to rewrite the derivative of ex as f ′ (x) = limh → 0exeh − ex h Factor ex out in the numerator f ′ (x) = limh → 0ex(eh − 1) h Since ex does not depend on h , the above may be rewritten as f ′ (x) = ex limh → 0eh − 1 h We now need to find the limit limh → 0eh − 1 h Let y = eh − 1

The derivative formula is defined for a variable 'x' having an exponent 'n' The exponent 'n' can be an integer or a rational fraction Hence, the formula to calculate the derivative is d dxxn = nxn−1 d d x x n = n x n − 1 That is, the derivative of the function ƒ(x) = e 2x is ƒ'(x) = 2e 2x This derivative tells us the rate of change the output of the original function per change in input Basically, the two equations tell us that the output of the function ƒ(x) = e 2x grows by a factor of 2e 2x per input So if our x value is one, plugging that value intoLet us Find a Derivative!

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The derivative of e x is e x This is one of the properties that makes the exponential function really important Now you can forget for a while the series expression for the exponential We only needed it here to prove the result above We can now apply that to calculate the derivative of other functions involving the exponential Example 1 f(x) = e axE^x times 1 f' (x)= e^ x this proves that the derivative (general slope formula) of f (x)= e^x is e^x, which is the function itself In other words, for every point on the graph of f (x)=e^x, the slope of the tangent is equal to the yvalue of tangent point So if y=Implicit\derivative\\frac{dy}{dx},\(xy)^2=xy1 \frac{\partial}{\partial y\partial x}(\sin (x^2y^2)) \frac{\partial }{\partial x}(\sin (x^2y^2))

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D d x (sech 1 x) = – 1 x 1 − x 2, (0 < x < 1) d d x (cosech 1 x) = – 1 x x 2 1, x ≠ 0 d d x (e ax sin bx) = e ax () d d x (e ax cos bx) = e ax () 5 Some theorems on DifferentiationDerivative Formula Assuming f is a realvalued function, then is called the derivative of at iff exists finitely Its derivative is said to be for function , provided that the above equation exists Here, search all the derivative formulas relating to trigonometric functions, inverse functions, hyperbolic functions, etc This means that the derivative of an exponential function is equal to the original exponential function multiplied by a constant ( k) that establishes proportionality d dx ax = kax d d x a x = k a x The proportionality constant is equal to the natural log of the base of the exponent d dx ax = ln(a)× ax d d x a x = ln

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Exponential functions and their corresponding inverse functions, called logarithmic functions, have the following differentiation formulas Note that the exponential function f ( x) = e x has the special property that its derivative is the function itself, f ′ ( x) = e x = f ( x ) Example 1 Find f ′ ( x) if Example 2 Find y ′ ifYou can obtain the formula for the derivative of any other base a > 0 by noting that y = a xis equal to elnax = e lna Use chain rule and the formula for derivative of ex to obtain that y0= exlna lna = ax lna Thus the derivative of a xis a lnaHandout Derivative Chain Rule PowerChain Rule a,b are constants Function Derivative y = a·xn dy dx = a·n·xn−1 Power Rule y = a·un dy dx = a·n·un−1 du dx PowerChain Rule Ex1a Find the derivative of y = 8(6x21)8 Answer y0 = 384(6x 21)7 a = 8, n = 8

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BASIC ANTIDERIVATIVE FORMULAS YOU REALLY NEED TO KNOW !! 1 Derivative Of f (x) = ax^n From the above calculations, we can conclude that the derivative of ax^n is anx^ (n1) For example, f (x) = 3x⁵, then the derivative — f' (x) — is 3×5x⁴=15x⁴ From this formula, we additionally understand that the derivative of a continuous is 0Derivatives are the fundamental tool used in calculus The derivative measures the steepness of the graph of a given function at some particular point on the graph Thus, the derivative is also measured as the slope It means it is a ratio of change in the value of the function to change in the independent variable

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It is the system we use in all theoretical work (In the next Lesson, we will see that e is approximately 2718) The system ofFormulas and examples of the derivatives of exponential functions, in calculus, are presented Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined Find the derivative of f(x) = e x / ( 1 x ) Solution to Example 3DERIVATIVE OF A TOWER OF EXPONENTS First, let us consider the derivative (with respect to x) of xˣ (which I will write as x^x) Let y = x^x Then y = e^ln (x)^x = e^ ln (x)*x, where e is the base of natural logarithms

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It is possible to write more accurate formulas than (53) for the first derivative For example, a more accurate approximation for the first derivative that is based on the values of the function at the points f(x−h) and f(xh) is the centered differencing formula f0(x) ≈ f(xh)−f(x−h) 2h (54)Given a function , there are many ways to denote the derivative of with respect to The most common ways are and When a derivative is taken times, the notation or is used These are called higherorder derivatives Note for secondorder derivatives, the notation is often used At a point , the derivative is defined to be Derivative of logₐx (for any positive base a≠1) Practice Derivatives of aˣ and logₐx Worked example Derivative of 7^ (x²x) using the chain rule Worked example Derivative of log₄ (x²x) using the chain rule Worked example Derivative of sec (3π/2x) using the chain rule Worked example Derivative of ∜ (x³4x²7) using the

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Ex dx = ex C ax dx = ax lna C 1 x dx =lnx C cosxdx=sinxC sec2 xdx=tanxC sinxdx= −cosx C csc2 xdx= −cotx C secxtanxdx=secx C 1 1x2 dx =arctanxC 1 √ 1− x2 dx =arcsinxC cscxcotxdx= −cscx C secxdx=lnsecxtanx C cscxdx= −lncscxcotx C xn dx = xn1 n1 C, when n = −1 Here areDerivative Formulas 1 0 dx d (the derivative of a constant is zero) 2 1 ( ) n n x nx dx d 3 2 1 1 dx x x d 4 x x dx d 2 1 ( ) 5 (u v) u' v' dx d 6 (C u) C u' dx d (C is a constant) 7 (uv) u'v uv' dx d (product rule) 8 (uvw) u'vw uv'w uvw' dx d (general product rule) 9 2 ' ' v u v uv v u dx d (quotient rule) 10 x x e e dx d ( ) 11Formula for the derivative of a specific function corresponds to a formula for the derivative of an elementary function The following table lists integration formulas side by side with the corresponding differentiation formulas Z xn dx = xn1 n1

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 This calculator finds the derivative of an entered function and tries to simplify the formula Use the "Function" field to enter a mathematical expression with x variable You can use operations like addition , subtraction , division /, multiplication *, power ^, and common mathematical functionsYou can find the full syntax description below the calculator$\large If\f(x)=e^{axb}\then\F(x)=\frac{1}{a}e^{axb}C$ $\large If\f(x)=\cos(axb)\then\F(x)=\frac{1}{a}\sin(axb)C$ $\large If\f(x)=\sin(axb)\then\F(x)=\frac{1}{a}\cos(axb)C$The Derivative Calculator supports computing first, second, , fifth derivatives as well as differentiating functions with many variables (partial derivatives), implicit differentiation and calculating roots/zeros You can also check your answers!

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The Derivative

Perhaps the most surprising and beautiful result in all of mathematics, Euler's formula,e^ix = cos(x) i sin(x), turns the theory of trigonometry into a si14 DERIVATIVES OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS The derivative of ln x The derivative of e with a functional exponent The derivative of ln u() The general power rule T HE SYSTEM OF NATURAL LOGARITHMS has the number called e as it base;These formulas can all be derived directly from the de nition of the derivative, with the except of ln(x), which requires a little extra work 2 Advanced building blocks Either memorize the derivatives of the following functions, or know how to derive them from the derivatives of the basic building blocks Either way,

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 We can write the derivative of this function in a ricorsive way Ricorsive way means that also in the definition there is the concept of derivate!To find the derivative of a function y = f(x) we use the slope formula Slope = Change in Y Change in X = ΔyΔx And (from the diagram) we see thatIf the base is equal to the number e a = e ≈ , then the derivative is given by d dx (ex) = (ex)′ = ex

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Consequently, the derivative of the logarithmic function has the form (logax)′ = 1 x logae By the changeofbase formula for logarithms, we have logae = lne lna = 1 lna Thus, y′(x) = (logax)′ = 1 xlna If a = e, we obtain the natural logarithm the derivative of which is expressed by the formulaThe constant e = 2718 is the unique basis for which the constant of proportionality is 1, so that the function is its own derivative This function, also denoted as exp x, is called the "natural exponential function", or simply "the exponential function" Since any exponential function can be written in terms of the natural exponential asOkay let's try this out on h of x equals e to the x squared plus 3x1 and let's observe that again the outside function is e to the x and the inside function is this polynomial x squared plus 3x1 and so the derivative according to this formula is the same function e to the g of x right so e to the x squared plus 3x1 times g prime of x and

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