Derivative of a Function

Let $f$ be a function. Suppose that the limit
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exists. The limit is denoted by $f^{\prime }(a)$ and is called the derivative of $f$ at $a$. In this case way say $f$ is differentiable at $a$.

Notice that to say the limit
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exists is the same as to say the limit
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exists. In this case the two limits are equal.

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If $0<x<\frac \pi 2,$ then MATH Hence
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Since MATH it follows that MATH

MATH

Since
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we have
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for $0<x<\frac \pi 2.$ Therefore
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for MATH Since MATH we have MATH

Derivative of $\cos \,x$

It follows from the identity


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that
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Therefore
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Derivative of $\sin \;x$

It follows from the identity
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that
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Therefore
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Higher-Order Derivatives

Let $f^{\prime }(x)$ be the derivative of the function $f;$ then the derivative of the function $f^{\prime }(x)$ is called the second derivative of the function $f(x)$ and is denoted by MATH A derivative of the second derivative is called the third derivative of the function $f(x).$ It is denoted by MATH Similarly we can define the fourth order $f^{IV}(x),$ fifth order $f^{V}(x)$ and so forth. A derivative of the nth order is symbolized by $f^{(n)}(x)$ and is sometimes also written MATH

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The above formulae show that the function $\sin \;x$ has derivatives of all orders and that
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This formula is sometimes written
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Exercise: Obtain the general formula for the nth order derivative of the function
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Let $f(x)=\frac 1{1+x}.$ Then
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In general we see that the nth order derivative takes the form
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Exercise: Find the general formula for the nth derivative of function
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The Coefficients of a Polynomial Function

Let $p(x)$ be a polynomial function of the form
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Then
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From
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we have
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Taking derivatives, we have
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therefore
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Taking derivatives again, we have
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therefore
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Continuing this way we see that the coefficients $c_{k}$ can be obtained from the formula
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In the same fashion, we see that if the polynomial $p(x)$ were expressed in the form
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then the coefficients $c_{k}$ can be obtained from the formula
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Linear Combinations of Functions of the Form $\frac 1{ax+b}$

Exercise: Apply the Expand command of Maple to express these functions as linear combinations of functions of the form $\frac{1}{ax+b}.$

(a) MATH

(b) MATH

(c) MATH

(d) MATH

Exercise: Find the nth derivative of each of the above functions.

Exercise: Find the nth order derivative of the function
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Leibniz Rule

These are the nth derivative of the product $uv$ of two functions $u=u(x),v=v(x):$
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In general, we have
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Rules for taking the Derivative

Suppose that $f^{\prime }(a)\;$and $g^{\prime }(a)$exist$.$ Then

(i) MATH

(ii) MATH

(iii) (MATH for scalars $\alpha ,\beta $$.$

If in addition that $g(x)\ne 0$ for all $x,$then

(iv) MATH

(v) MATH

(vi) Let $p$ be a polynomial function. Then MATH

Exercise: prove each of the above statements.

Chain Rule

If $g$ is differentiable at $a$ and $f$ is differentiable at $g(a),$ then the composition $f\circ g$ is differentiable at $a,$ and
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Proof. Define a function $\phi $ as follows.
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We claim that $\phi $ is continuous at $h=0.$

Let $\epsilon >0$ be given. Since $f$ is differentiable at $g(a),$ there exists MATH such that
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Since $g$ is differentiable at $a,$ it follows that $g$ is continuous at $a,$ so there exists $\delta >0$ such that
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Now fix any $h$ with MATHWe show that MATH

Case 1): $g(a+h)-g(a)\ne 0.$ Then
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Write $k=g(a+h)-g(a).$ Then $k\ne 0$ and MATH and so
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therefore MATH

Case 2): $g(a+h)-g(a)=0.$ Then MATH so clearly MATH

We have proved that
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Now for $h\neq 0$ we have
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even if $g(a+h)-g(a)=0$ (both sides are zero!) Therefore
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