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Finding Derivatives of Basic Functions

Quick Overview

Examples

Example 1

Find ddx(8x3)

Step 1

Use the rule for linear functions.

We know that the derivative of any linear function is just the slope of the line. Consequently,

ddx(8x3)=8

Example 2

Suppose f(x)=sin3x4cos7x. Find f(π2)

Step 1

Find the f(x) using the rules (not the definition).

f(x)=ddx(sin3x)ddx(4cos7x)(Difference Rule)=ddx(sin3x)4ddx(cos7x)(Coefficient Rule)=3cos3x4ddx(cos7x)(Derivative of the Sine)=3cos3x4(7sin7x)(Derivative of the Cosine)=3cos3x+28sin7x

Step 2

Evaluate the derivative at x=π/2.

f(π2)=3cos(3π2)+28sin(7π2)=3cos(3π2)+28sin(7π2)=3(0)+28(1)=28

Answer

f(π2)=28 when f(x)=sin3x4cos7x.

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