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
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
1. Differentiate cos³x with respect to x.
Let y = cos³x
Let u = cos x
therefore y = u³
dy = 3u²
du = -sin x
dy = du × dy
dx dx du
= -sin x × 3u²
= -sin x × 3cos²x
= -3cos²x sin x
Differentiation of exponential functions
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:
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
We let u = ln 2x and v = e2x + 2, and we'll use the derivative of a quotient formula
u = ln 2x = ln 2 + ln x
And for v = e2x + 2 we have:
Apply Quotient rule:
Using the derivatives we just found for u and v gives:
6. Find the derivative of
7. Find the derivative of
then y = u3.
4. Calculate the gradient(s) of the curve at the point(s) where it crosses the given line.
y = 3x2 - 2x + 6, y-axis.
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