Back to Differentiation: Definition and Basic Derivative Rules

Lesson PreviewLesson 7 of 10 in this unit

2.7 Derivatives of cos x, sin x, eˣ, and ln x

Part of Differentiation: Definition and Basic Derivative Rules in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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Some functions are so important in calculus that their derivatives are treated as core facts to know immediately. In this lesson, the key results are
ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x)=\cos x
,
ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x)=-\sin x
,
ddx(ex)=ex\frac{d}{dx}(e^x)=e^x
,
and
ddx(ln⁡x)=1x\frac{d}{dx}(\ln x)=\frac{1}{x}
for x>0.

These rules matter because trigonometric, exponential, and logarithmic functions appear constantly in modeling, motion, growth, decay, and later integration work. Unlike the power rule, these derivatives do not come from just lowering an exponent, so they must be learned as their own fundamental derivative relationships.

In this unit

10 lessons

  1. 2.1Defining Average and Instantaneous Rates of Change at a PointPreview lesson
  2. 2.2Defining the Derivative of a Function and Using Derivative NotationPreview lesson
  3. 2.3Estimating Derivatives of a Function at a PointPreview lesson
  4. 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not ExistPreview lesson
  5. 2.5Applying the Power RulePreview lesson
  6. 2.6Derivative Rules: Constant, Sum, Difference, and Constant MultiplePreview lesson
  7. 2.7Derivatives of cos x, sin x, eˣ, and ln xThis lesson
  8. 2.8The Product RulePreview lesson
  9. 2.9The Quotient RulePreview lesson
  10. 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant FunctionsPreview lesson

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