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Differentiation

3.4 Differentiation Rules for Circular Functions

Differentiation Rules for \sin(x), \cos(x), \tan(x)

  • The other rules regarding differentiation such as addition, subtraction, multiplication, division of functions, chain rule etc. applies too.
  • All of these rules can be proven via first principles, but it is not required.
  • The basic rules in this section are:

i) If f(x)=\sin(x), then f^{\prime}(x)=\cos x

ii) If f(x)=\cos x, then f^{\prime}(x)=-\sin x

iii) If f(x)=\tan x, than f^{\prime}(x)=\sec^2x=\sec x \times \sec x

  • It is possible to expand these further using chain rule:
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3.3 Differentiation Rules for Exponentials and Logarithms

Differentiation Rules for Exponentials and Logarithms

  • The other rules regarding differentiation such as addition, subtraction, multiplication, division of functions, chain rule etc. applies too.
  • All of these rules can be proven via first principles, but it is not required.
  • The basic rules in this section are:

i) If f(x)=a^{x}, \text { then } f^{\prime}(x)=a^{x} \ln a

ii) If f(x)=\log _{a} x, \text { then } f^{\prime}(x)=\frac{1}{x}(\ln a)^{-1}

where n \neq 0, n \in R (or n \in R \backslash\{0\})

  • Applying the rules to natural exponential and logarithm functions (i.e. a=e), yields:
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3.2 Finding Gradient of a Tangent Line

Finding Gradient of a Tangent Line

  • Recall that the tangent line to f(x) at point P(a,\ f(a)) has gradient f^{\prime}(a). Using these rules, we can find gradient of tangents of polynomial functions easily and quickly.

Example

Find the gradient of tangent line to the curve f(x)=3x^3-6x^2+1 at point (1,\ -2).

f^{\prime}(x)=9x^2-12x, so the gradient when x=1 is f^{\prime}(1)=9 \times 1^{2}-12 \times 1=-3

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3.1 Differentiation Rules for Polynomials

Differentiation Rules for Polynomials

  • The other rules regarding differentiation such as addition, subtraction, multiplication, division of functions, chain rule etc. applies too.
  • All of these rules can be proven via first principles, but it is not required.
  • The only rule for differentiating polynomials:

\text { If } f(x)=x^{n}, \text { then } f^{\prime}(x)=n x^{n-1}

where n \neq 0, n \in R (or n \in R \backslash\{0\})

Note: If n=0, then f(x)=a \text { constant, } f^{\prime}(x)=0

Proof (Not required)

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1.2 Derivatives Basic Concepts – Tangent and Derivatives

Chord, Secant, and Tangent

  • A chord is a line segment joining two points on a curve, like the line AB in the previous example.
  • A secant is a line that passes through two points, which means that it can be longer than a chord for any given two points A and B.
Picture 1
  • Graphically, ‘a tangent though a curve at point P’ is defined as a line that passes through P and its gradient equals the curve’s instantaneous rate of change at P.
  • Suppose P has coordinates P(p,\:f(p)), then \text {Gradient of tangent at } P=\lim _{h \rightarrow 0} \frac{f(p+h)-f(p)}{h}

Note: Here, h \rightarrow 0 is used and not h \rightarrow 0^{+}.

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