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ACM000091

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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