Question

1.(5sin(x)-4x+x^6)^' = 2.(ln* )= 3.()^'= 4.()^'=

Answer 1

1. =5sinx-4x+x⁶

2. =ln(x¹/³)

1/3*ln(x)

3. =∛x

4. =3x+2/x²-1

Answer 2

To differentiate the given expressions, let's go through them one by one:

1. Differentiation of (5sin(x) - 4x + x^6): To differentiate this expression, we can differentiate each term separately using the power rule and the derivative of sine:

The derivative of 5sin(x) is 5cos(x). The derivative of -4x is -4. The derivative of x^6 is 6x^5.

Therefore, the derivative of (5sin(x) - 4x + x^6) is: 5cos(x) - 4 + 6x^5.

2. Differentiation of ln(x): The derivative of ln(x) is 1/x. Therefore, the derivative of ln(x) is 1/x.

3. Differentiation of (f(x))^n: To differentiate (f(x))^n, we can use the chain rule. The chain rule states that if we have a function raised to a constant power, we multiply the constant power by the derivative of the function itself and then multiply it by the derivative of the function inside.

So, if we have (f(x))^n, the derivative would be: n(f(x))^(n-1) * f'(x).

4. Differentiation of (a^x): To differentiate (a^x), we can use the natural logarithm. The derivative of a^x is given by a^x * ln(a).

Therefore, the derivative of (a^x) is: a^x * ln(a).

Please note that in expressions 3 and 4, you need to specify the function inside the parentheses or the value of 'a' for a more accurate differentiation result.