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Differentiation



Basic Rules

  • For a curve y = f(x), dy/dx represents the gradient function of the tangent to the curve at a point x. 
  • dy/dx measures the rate of change of y with respect to x.
  • The derivative of f(x) = x r where r is a constant real number is given by

    f '(x) = r x r - 1
  • Example
    • f(x) = 3x 3 ,

        f '(x) = 9 x 2
A. Sum Rule

The derivative of f(x) = g(x) + h(x) is given by

f '(x) = g '(x) + h '(x)

B. Difference Rule


The derivative of f(x) = g(x) - h(x) is given by

f '(x) = g '(x) - h '(x)

C. Product Rule



D. Quotient Rule



E. Chain Rule


F. More complex differentiation:

Differentiation of trigo functions

Examples

1. Differentiate cos³x with respect to x.

Let y = cos³x
Let u = cos x
therefore y = u³


dy   =  3u²
du

du  =  -sin x
dx

dy  =  du  ×  dy
dx      dx       du


     =  -sin x × 3u²
     = -sin x × 3cos²x
     = -3cos²x sin x

Differentiation of exponential functions



Examples

1. Differentiate y = ln2x

--> 2/2x = 1/x

2. Differentiate y = lnx2

y = lnx2 = 2lnx
dy/dx = 2/x

3. Differentiate y = 2 ln (3x2 − 1)

dy/dx = 12/(3x2 - 1)

4. Differentiate y = ln(1 2x)3 = 3ln(1 - 2x)

dy/dx = -6/(1 - 2x)

If u is a function of x, we can obtain the derivative of an expression in the form eu:


Examples

1. Find the derivative of y = 103x:
2. Find the derivative of y = ex2:
3. Find the derivative of y = sin(e3x).
4. Find the derivative of y = esin x.

5. Find the derivative of

math expression

We let u = ln 2x and v = e2x + 2, and we'll use the derivative of a quotient formula

math expression

u = ln 2x = ln 2 + ln x

\large{\frac{du}{dx}=\frac{1}{x}}

And for v = e2x + 2 we have:

\large{\frac{dv}{dx}=2e^{2x}}

Apply Quotient rule:

math expression

Using the derivatives we just found for u and v gives:

math expression

simplify...

Finally...math expression


6. Find the derivative of

math expression

Answer:

7. Find the derivative of

math expression

Answer:

Let

then y = u3.

So

and

So

Questions

1.

Answers:

2.


Answer:

3.
Answer:
19/27


4. Calculate the gradient(s) of the curve at the point(s) where it crosses the given line.
y = 3x2 - 2x + 6, y-axis.


Answer: -2


5. The gradient of the curve y = 2x2 + px + q at the point (1, 3) is 9. Calculate the values of p and q.


Answer: p = 5, q = -4
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