Differentiate
e^(2x^3+x) X tan(x^2)
tan e^x / ln(x^6/2-3x+3e)
cos(tan(e^4x^2))
To differentiate each of these expressions, we will use the basic rules of differentiation.
1. Differentiation of e^(2x^3+x) * tan(x^2):
To differentiate this product function, we will need to apply the product rule. The product rule states that if we have two functions u(x) and v(x), their product can be differentiated using the formula (u * v)' = u'v + uv'.
Let's break down the given expression:
u(x) = e^(2x^3+x)
v(x) = tan(x^2)
Now, we need to find the derivatives of u(x) and v(x) individually and then apply the product rule:
Derivative of u(x):
u'(x) = (2x^3+x)' * e^(2x^3+x)
= (6x^2 + 1) * e^(2x^3+x)
Derivative of v(x):
v'(x) = tan(x^2)'
= (sec^2(x^2)) * (x^2)'
= 2x * sec^2(x^2)
Now, applying the product rule:
(u * v)' = u'v + uv'
= ((6x^2 + 1) * e^(2x^3+x)) * tan(x^2) + e^(2x^3+x) * (2x * sec^2(x^2))
2. Differentiation of tan(e^x / ln(x^6/2-3x+3e)):
To differentiate this quotient function, we will use the quotient rule. The quotient rule states that if we have two functions u(x) and v(x), their quotient can be differentiated using the formula (u / v)' = (u'v - uv') / v^2.
Let's break down the given expression:
u(x) = tan(e^x)
v(x) = ln(x^6/2-3x+3e)
Now, we need to find the derivatives of u(x) and v(x) individually and then apply the quotient rule:
Derivative of u(x):
u'(x) = (tan(e^x))'
= sec^2(e^x) * (e^x)'
Derivative of v(x):
v'(x) = (ln(x^6/2-3x+3e))'
= (1 / (x^6/2-3x+3e)) * (x^6/2-3x+3e)'
Now, applying the quotient rule:
(u / v)' = ((sec^2(e^x) * (e^x)) * ln(x^6/2-3x+3e) - tan(e^x) * (1 / (x^6/2-3x+3e)) * (x^6/2-3x+3e)') / (ln(x^6/2-3x+3e))^2
3. Differentiation of cos(tan(e^4x^2)):
To differentiate this composition of functions, we will use the chain rule. The chain rule states that if we have a function f(g(x)), the derivative of f(g(x)) is given by (f(g(x)))' = f'(g(x)) * g'(x).
Let's break down the given expression:
f(x) = cos(x)
g(x) = tan(e^4x^2)
Now, we need to find the derivatives of f(x) and g(x) individually and then apply the chain rule:
Derivative of f(x):
f'(x) = cos(x)'
Derivative of g(x):
g'(x) = (tan(e^4x^2))'
Now, applying the chain rule:
(cos(tan(e^4x^2)))' = f'(g(x)) * g'(x)
= (-sin(g(x))) * (tan(e^4x^2))'
Remember to simplify or substitute the appropriate values for g'(x) and (tan(e^4x^2))' based on the specific derivatives you are seeking for each expression.