(i) To differentiate em cos-1(x sin(nx)), we can use the chain rule. Let's break it down:
First, let's differentiate the outer function, which is em cos-1(u), where u = x sin(nx). The derivative of em cos-1(u) is em cos-1(u) * (-1 / sqrt(1 - u²)).
Next, we need to differentiate the inner function, u = x sin(nx). The derivative of u with respect to x is sin(nx) + x cos(nx) * n.
Combining the derivatives using the chain rule, the overall derivative is:
em cos-1(u) * (-1 / sqrt(1 - u²)) * (sin(nx) + x cos(nx) * n)
(ii) To differentiate -tan(x) * sqrt(1 + x²), we can use the product rule. Let's break it down:
The derivative of -tan(x) with respect to x is -sec²(x).
The derivative of sqrt(1 + x²) with respect to x is (1/2) * (1 + x²)^(-1/2) * (2x).
Using the product rule, the overall derivative is:
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Step-by-step explanation:
Let's differentiate each expression step by step:
(i) To differentiate em cos-1(x sin(nx)), we can use the chain rule. Let's break it down:
First, let's differentiate the outer function, which is em cos-1(u), where u = x sin(nx). The derivative of em cos-1(u) is em cos-1(u) * (-1 / sqrt(1 - u²)).
Next, we need to differentiate the inner function, u = x sin(nx). The derivative of u with respect to x is sin(nx) + x cos(nx) * n.
Combining the derivatives using the chain rule, the overall derivative is:
em cos-1(u) * (-1 / sqrt(1 - u²)) * (sin(nx) + x cos(nx) * n)
(ii) To differentiate -tan(x) * sqrt(1 + x²), we can use the product rule. Let's break it down:
The derivative of -tan(x) with respect to x is -sec²(x).
The derivative of sqrt(1 + x²) with respect to x is (1/2) * (1 + x²)^(-1/2) * (2x).
Using the product rule, the overall derivative is:
(-tan(x)) * (1/2) * (1 + x²)^(-1/2) * (2x) + (-sec²(x)) * sqrt(1 + x²)
(iii) To differentiate √x (2x+3)² / (x+1), we can use the quotient rule. Let's break it down:
The derivative of √x with respect to x is (1/2) * x^(-1/2).
The derivative of (2x+3)² with respect to x is 2 * (2x+3) * (2).
The derivative of (x+1) with respect to x is 1.
Using the quotient rule, the overall derivative is:
[(x+1) * 2 * (2x+3) * (2) - √x * 2 * (2x+3)] / (x+1)²
(iv) To differentiate (x-3) (x²+4) / (3x²+4x+5), we can again use the quotient rule. Let's break it down:
The derivative of (x-3) (x²+4) with respect to x is (x²+4) + (x-3) * (2x).
The derivative of (3x²+4x+5) with respect to x is 6x + 4.
Using the quotient rule, the overall derivative is:
[(3x²+4x+5) * (x²+4) - (x-3) * (6x+4)] / (3x²+4x+5)²
These are the derivatives of the given expressions.