Derivative Step by Step

erivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity.-  Source : Wikipedia

Calculus is used to solve differentiation of any subjective equation and the equivalent output result. we need to find f(x) where,     d/dx f(x) = g(x).The derivative of constant is zero in calculus. The linear equations are also compared using calculus. Integration is considered as one of most important study of calculus in mathematics. British mathematician, Isaac Newton and the German man Gottfried Leibnitz, invented the calculus. The methods that are applied in continous graphs, curves or fucntions  is calculus.

Understanding Derivative of Cot x is always challenging for me but thanks to all math help websites to help me out.

Steps to solve the Derivative function:

Derivative  problem 1:

Find the first derivative and second derivative of

f(x) = x4 + 7x3 - 3x2 - 9x +3

Solution:

Step 1:  First differentiate f(x) = x4 + 7x3 - 3x2 - 9x +3

Step 2:  when we differentiate the first term we get  (1 × 4) x (4-1)

Step 3:  The above equation can above simplified as  4 x 3

Step 4:  Like wise we can differentiate  and simplify all the terms in the equation

Step 5:  Finally after the first derivative we will get  4 x 3 + 21 x2 - 6x - 9

Step 6:  Now we will go for Second derivative
End of the second derivative we will get   12x2 + 42 x - 6


First derivative:

f' = (df )/ (dx)

= (1 × 4) x (4-1) + (7 × 3)x(3-1) - (3 × 2)x(2-1) - (9 × 1)x (1-1)

= 4 x 3 + 21 x2 – 6x - 9

Second derivative:

f '' = (df ' )/ (dx)

= 4 x 3 + 21 x2 – 6x - 9

= 12x2 + 42 x - 6

Answer:

(d^2)/(dx^2)  (x4 + 7x3 - 3x2 - 9x +3) = 12x2 + 42 x - 6

Derivative problem 2:

Find the differential for y = sqrt(2x^3 - 9x)

Solution:

Step 1:  First turn sqrt(2x^3 - 9x)to (2x3 - 9x)1/2
Step 2:  Differentiate in the usual method
Step 3:  Then finally we get,

d/(dx) ((2x3 - 9x)1/2 ) = ½ ( 2x3 - 9x ) ½ -1( 6x2 - 9 )

= ½ ( 2x3 - 9x ) -1/2( 6x2 - 9 )

= ( 6x^2 - 9 ) / (2 sqrt (2x^3 - 9x ))

Answer of given calculus test exam problem is

d/(dx) ((2x3 - 9x)1/2 ) = ( 6x^2 - 9 ) / (2 sqrt (2x^3 - 9x ))

My forthcoming post is on how to do algebra 2 problems step by step and sample papers for class 9 cbse will give you more understanding about Algebra.

Derivative practice problem:

Derivative practice problem 1:

Find the derivative of f(x) = ( x - 5 ) ( 2x + 1 )

Answer:   d/(dx) f(x) = 4x - 9

Derivative practice problem 2:

Differentiate the given equation with respect to t. y =  8t3 + 12t

Answer:  24t2   + 12