Author: misamaliraza94
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Derivatives of Elementary Functions
The three basic derivatives of the algebraic, logarithmic/ exponential and trigonometric functions are derived from the fundamental principle of differentiation and are used as standard derivative formulas. They are as follows: If y = lnee x, then dy/dx = 1/x if y = logaa x, then dy/dx = 1/[(log a) x] If y = a x , dy/dx= ax log a
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How To Find The Derivatives?
Derivatives are obtained by applying the limits as per the first principle of differentiation that we obtained as the definition of a derivative. Let f(x) = 4×2 + 3 f′(x)=lim δx→0 f(x+δx) − f(x) / δx f ′ ( x ) = lim δ x → 0 f ( x + δ x ) − f ( x…
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Interpretation of Derivatives
The derivative is a function that is geometrically defined as the slope of the line tangent to the curve at any point. If f is differentiable and continuous at [a,b], then This change in h is infinitely very small. We denote it by Δx. Then change in the original function f(x) is also small, denoted…
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Derivatives
A derivative is the rate of change of a quantity y with respect to another quantity x. A derivative is also termed the differential coefficient of y with respect to x. Differentiation is the process of finding the derivative of a function. If f(x) is a function differentiable in an interval [a,b], at every point of the…
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Limits and Functions
A function may approach two different limits. One where the variable approaches its limit through values larger than the limit and the other where the variable approaches its limit through values smaller than the limit. In such a case, the limit is not defined but the right and left-hand limits exist. When the limx→af(x)=A+limx→af(x)=A+ given the values of f…
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Special Rules:
1. limx→axn−anx−a=na(n−1)limx→axn−anx−a=na(n−1), for all real values of n.2. limθ→0sinθ/θ=1 limθ→0 sinθ/θ=13. limθ→0tanθ/ θ=1 limθ→0tanθ/θ=14. limθ→01−cosθ/θ=0 limθ→01−cosθ/θ=05. limθ→0 cosθ=1 limθ→0 cosθ=16. limx→0ex=1limx→0ex=17. limx→0ex−1x=1limx→0ex−1x=18. limx→∞(1+1x)x=e
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Properties of Limits
Here are some properties of the limits of the function: If limits limx→alimx→a f(x) and limx→alimx→a g(x) exists, and n is an integer, then, Law of Addition: limx→a[f(x)+g(x)]=limx→af(x)+limx→ag(x)limx→a[f(x)+g(x)]=limx→af(x)+limx→ag(x) Law of Subtraction: limx→a[f(x)−g(x)]=limx→af(x)−limx→ag(x)limx→a[f(x)−g(x)]=limx→af(x)−limx→ag(x) Law of Multiplication: limx→a[f(x)⋅g(x)]=limx→af(x)⋅limx→ag(x)limx→a[f(x)⋅g(x)]=limx→af(x)⋅limx→ag(x) Law of Division: limx→a[f(x)g(x)]=limx→af(x)limx→ag(x), where limx→ag(x)≠0limx→a[f(x)g(x)]=limx→af(x)limx→ag(x), where limx→ag(x)≠0 Law of Power: limx→ac=c
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Limits and Functions
A function may approach two different limits. One where the variable approaches its limit through values larger than the limit and the other where the variable approaches its limit through values smaller than the limit. In such a case, the limit is not defined but the right and left-hand limits exist. When the limx→af(x)=A+limx→af(x)=A+ given the values of f…
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What Are Limits?
Limits in maths are unique real numbers. Let us consider a real-valued function “f” and the real number “c”, the limit is normally defined as limx→cf(x)=Llimx→cf(x)=L. It is read as “the limit of f of x, as x approaches c equals L”. The “lim” shows the limit, and fact that function f(x) approaches the limit L as x…
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Important Notes on Rational Function
A rational function equation is of the form f(x) = P(x) / Q(x), where Q(x) ≠ 0. Every rational function has at least one vertical asymptote. Every rational function has at most one horizontal asymptote. Every rational function has at most one slant asymptote. The excluded values of the domain of a rational function help…