Pure Mathematics · 3rd secondary · First Term

Calculus - Differentiation and its applications — Derivatives of the exponential and logarithmic functions

About this lesson

This unit covers the differentiation of exponential and logarithmic functions, including the natural exponential function e^x and the natural logarithmic function ln x. It explains their definitions, domains, ranges, and key properties, and then derives rules for their derivatives. The unit also introduces the chain rule for composite functions, derivatives of exponential functions with base a, derivatives of logarithmic functions with base a, and logarithmic differentiation as a technique for differentiating functions of the form f(x)^g(x). Applications include finding slopes of tangents and normals, solving problems involving related rates, and proving identities involving derivatives.

Main topics in this lesson

  • Natural exponential function and its derivative
  • Natural logarithmic function and its derivative
  • Chain rule for exponential and logarithmic functions
  • Derivatives of exponential functions with base a
  • Derivatives of logarithmic functions with base a
  • Logarithmic differentiation
  • Applications: tangents, normals, and modeling problems

Key terms and vocabulary

  • natural exponential function
  • natural logarithmic function
  • derivative
  • chain rule
  • logarithmic differentiation
  • base e
  • base a
  • tangent
  • normal
  • slope

Questions and answers on Calculus - Differentiation and its applications — Derivatives of the exponential and logarithmic functions - practice and revision

Multiple choice questions on this lesson in Pure Mathematics for 3rd secondary, First Term, with practice exercises for revision and exam preparation.

1

إذا كانت ص = ه^(لوق س) حيث س > ٠ فإن صَ تساوي

  • 1 ١
  • 2 س
  • 3 ه^(س)
  • 4 ١/س
2

إذا كانت ص = لوق(س² + ١) فإن صَ تساوي

  • 1 ٢س/(س² + ١)
  • 2 ١/(س² + ١)
  • 3 ٢س(س² + ١)
  • 4 ٢/(س² + ١)
3

إذا كانت ص = ه^(٣س) فإن المشتقة الثانية صََ تساوي

  • 1 ٩ ه^(٣س)
  • 2 ٣ ه^(٣س)
  • 3 ٦ ه^(٣س)
  • 4 ه^(٣س)
4

إذا كانت ص = ه^(س) فإن ميل العمودي على المنحنى عند النقطة (٠ ، ١) يساوي

  • 1 −١
  • 2 ١
  • 3 ٠
  • 4 ه
5

إذا كان ميل المماس للمنحنى ص = ه^(أس) عند س = ٠ يساوي ٣ فإن قيمة أ تساوي

  • 1 ٣
  • 2 ١
  • 3 ٩
  • 4 ١/٣
6

ميل المماس للمنحنى ص = لوق س عند س = ١ يساوي

  • 1 ١
  • 2 ٠
  • 3 ه
  • 4 −١

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